(navigation image)
Home American Libraries | Canadian Libraries | Universal Library | Community Texts | Project Gutenberg | Children's Library | Biodiversity Heritage Library | Additional Collections
Search: Advanced Search
Anonymous User (login or join us)
Upload
See other formats

Full text of "The Collected Mathematical Papers of Arthur Cayley"

Google 



This is a digital copy of a book that was preserved for generations on library shelves before it was carefully scanned by Google as part of a project 

to make the world's books discoverable online. 

It has survived long enough for the copyright to expire and the book to enter the public domain. A public domain book is one that was never subject 

to copyright or whose legal copyright term has expired. Whether a book is in the public domain may vary country to country. Public domain books 

are our gateways to the past, representing a wealth of history, culture and knowledge that's often difficult to discover. 

Marks, notations and other maiginalia present in the original volume will appear in this file - a reminder of this book's long journey from the 

publisher to a library and finally to you. 

Usage guidelines 

Google is proud to partner with libraries to digitize public domain materials and make them widely accessible. Public domain books belong to the 
public and we are merely their custodians. Nevertheless, this work is expensive, so in order to keep providing tliis resource, we liave taken steps to 
prevent abuse by commercial parties, including placing technical restrictions on automated querying. 
We also ask that you: 

+ Make non-commercial use of the files We designed Google Book Search for use by individuals, and we request that you use these files for 
personal, non-commercial purposes. 

+ Refrain fivm automated querying Do not send automated queries of any sort to Google's system: If you are conducting research on machine 
translation, optical character recognition or other areas where access to a large amount of text is helpful, please contact us. We encourage the 
use of public domain materials for these purposes and may be able to help. 

+ Maintain attributionTht GoogXt "watermark" you see on each file is essential for in forming people about this project and helping them find 
additional materials through Google Book Search. Please do not remove it. 

+ Keep it legal Whatever your use, remember that you are responsible for ensuring that what you are doing is legal. Do not assume that just 
because we believe a book is in the public domain for users in the United States, that the work is also in the public domain for users in other 
countries. Whether a book is still in copyright varies from country to country, and we can't offer guidance on whether any specific use of 
any specific book is allowed. Please do not assume that a book's appearance in Google Book Search means it can be used in any manner 
anywhere in the world. Copyright infringement liabili^ can be quite severe. 

About Google Book Search 

Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers 
discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web 

at |http: //books .google .com/I 



)RD UNIVERSITY LIBKAKII S STANFORD UNIVLlJSirr LIUKAUHS STA 



SITY LIBRARIES . STANFORD UNIVERSITY LIBRARIES ■ STANFORD (JNI 



lES STANFORD UNIVERSITY LIBRARIES STANFORD UNIVERSITY LI Bl 



STANFORD university libraries STANFORD UNIVERSR 



STANFORD UNIVERSITY LIBRARIES STANFORD UNIVERSITY LIBRARIE 



LIBRARIES STANFORD UNIVERSITY LIBRARIES STANFOf 



)RD university LIBRARIES STANFORD UNIVERSITY LIBRARIES -STA 



\SITY LIBRARIES . STANFORD UNIVERSITY LIBRARIES . STANFORD UNI 



I^S ■ STANFORD UNIVERSITY LIBRARIES ■ STANFORD UNIVERSITY LIBI 



IBRARIES STANFORD university LIBRARIES STANFORD UNIVERS 



TANFORD UNIVERSITY LIBRARIES STANFORD (JN IVERSITY LIBRARIE 



LIBRARIES STANFORD UNIVERSITY LIBRARIES STANFOf 



^RD UNIVERSITY LIBRARIES STANFORD UNIVERSITY LIBRARIES STA 



UNIVERSITY L'BR 



iry LinHAWUS STANFORD UNIVERSITV LIBRARIES STANFORD UNIVER 



ARIES . STANFORD UNIVERSITY LIBRARIES . STANFORD UNIVERSITY 



FORD UNIVEHSirf LIBRARIES STANFORD UNIVERSITY LIBRARIES 



iTANFORD UNIVERSITY LIBRARIES STANFORD UNIVERSITY LIBRARIES ST 



UNIVERSITY LIBRARIES . STANFORD UNIVERSITY LIBRARIES STANFORD Uf 



-IBRARIES ■ STANFORD UNIVERSITY LIBRARIES ■ STANFORD UNIVERSITY [_([ 



EHSITY LIBRARIES STANFORD UNIVERSITY LIBRARIES STANFORD UNIVER ' 



BAHIES STANFORD UNIVERSITY LIBRARIES ■ STANFORD UNIVERSITY 



•"FORD UNIVERSITY LIBRARIES STANFORD UNIVERSITY LIBRARIES 



STANFORD 



LIBRARIES STANFORD UNIVERSITY LIBRARIES ST 



UNIVERSITY LIBRARIES . STANFORD UNIVERSITY LIBRARIES . STANFORD (JN 



-IBRARIES • STANFORD UNIVERSITY LIBRARIES STANF 



LIBRARIES STANFORD UNIVERSITY LIBRARIES STANFORD 



STANFORD UNIVERSITY LIBRARIES . STANFORD UNIVERSITY LIBRO 



-'""o University LIBRARIES Stanford university LIBP""'ES 



MATHEMATICAL PAPEKS. 



lonlion: C. J. CLAY & SONS, 

CAMBBIDGE UNIVERSITY PRESS WAREHOUSE, 

AVE MARIA LANE. 




Camfmlige: DEIGHTON, BELL AND CO. 
!Lrqj}ts : F. A. BROCEHAUS. 



THE COLLECTED 



MATHEMATICAL PAPERS 



OF 



AETHUE CAYLEY, Sc.D., F.E.S., 

8ADLERIAN PROFESSOR OF PURE MATHEMATICS IN THE UNIVERSITY OF CAMBRIDGE. 



VOL. II. 



CAMBRIDGE : 
AT THE UNIVERSITY PRESS. 

1889 

[All Rights reserved.^ 




9y^i_\o^^ 



CAMBRIDGE : 

PRINTED BY C. J. CLAT, ILA. AMD SONS, 
AT THE UNIVEBSITY PRESS. 



ADVEETISEMENT. 



THE present volume contains fifty-eight papers (numbered 101, 102,..., 158) 
originally published, all but two of them, in the years 1851 to 1860: 
they are here reproduced nearly but not exactly in chronological order. 
The two excepted papers are 142, Numerical Tables Supplementary to 
Second Memoir on Quantics, now first published (1889); and, 143, Tables 
of the Covariants M to W of the Binary Quintic : from the Second, 
Third, Fourth, Fifth, Eighth, Ninth and Tenth Memoirs on Quantics 
(arranged in the present form, 1889) : the determination of the finite 
number, 23, of the covariants of the quintic was made by Gordan in the 
year 1869, and the calculation of them having been completed in my 
Ninth and Tenth Memoirs, it appeared to me convenient in the present 
republication to unite together the values of all the covariants : viz. those 
of A to L are given in the Second Memoir 141, and the remainder 
M to W in the paper 143. I have added to the Third Memoir 144, in the 
notation thereof, some formulae which on account of a difference of notation 
were omitted fi:om a former paper, 35. 

I remark that the present volume comprises the first six of the ten 
Memoirs on Quantics, viz. these are 139, 141, 144, 155, 156 and 158. 
I have, in the Notes and References, inserted a discussion of some length 
in reference to the paper 121, Note on a Question in the Theory of 
Probabilities: and also some remarks in reference to the theory of Dis- 
tance developed in the Sixth Memoir on Quantics, 158. 



C. II. 



CONTENTS. 



PAOB 



101. Notes on Lagrange's Theorem ....... 1 

Gamb. and DubL Math. Jour. t. vi. (1851), pp. 37 — 45 

102. On a Double Infinite Series ....... 8 

Camb. and DubL Math. Jour, t vi. (1851), pp. 45 — 47 

103. On Certain Definite Integrals 11 

Camb. and DubL Math. Jour, t vl (1851), pp. 136—140 

104. On the Theory of Permutants 16 

Camb. and DubL Math. Jour, t vii. (1852), pp. 40 — 51 

105. Correction to the Postscript to the Pa/per on Permutants 27 

Camb. and DubL Math. Jour, t vii. (1852), pp. 97—98 

106. On the Singularities of Surfaces 28 

Camb. and DubL Math. Jour. t. vii. (1852), pp. 166—171 

107. On the Theory of Skew Surfaces 33 

Camb. and DubL Math. Jour, t vii. (1852), pp. 171—173 

108. On certain Multiple Integrals connected with the Theory of 

Attractions 35 

Camb. and DubL Math. Jour. t. vii. (1852), pp. 174—178 

109. On the Rationalisation of certain Algebraical Equ/Uions . 40 

Camb. and DubL Math. Jour. t. viii. (1853), pp. 97—101 

110. Note on the Transformation of a TrigonomstHcal Expression . 45 

Camb. and DubL Math. Jour. t. ix. (1854), pp. 61 — 62 

111. On a Theorem of M. Lejeune-Dirichlet' s 47 

Camb. and DubL Math. Jour. t. ix. (1854), pp. 163—165 

&2 



VUl CONTENTS. 



PAOB 



112. Demonstration of a Theorem relating to the Products of Sums 

of Squares 49 

Phil. Mag. t IV. (1852), pp. 516—519 

113. On the Geometrical Representation of the Integral 

Idx-^Jix + a) {x + h) {x + c) 53 

Phil. Mag. t. V. (1853), pp. 281—284 

114. Analytical Researches connected with Steiner's Extension of 

MalfattHs Problem 57 

Phil. Trans, t cxlii. (for 1852), pp. 253—278 

115. Note on the Porism of the In-and-circum^crihed Polygon . 87 

Phil. Mag. t. VI. (1853), pp. 99—102 

116. Correction of two Theorems rekuing to the In-and-circum^ 

scribed Polygon ......... 91 

Phil. Mag. t VI. (1853), pp. 376—377 

117. Note on the Integral \dx'^J{m'-x){x + a){x + h){x + c) . . 93 

Phil. Mag. t VL (1853), pp. 103—105 

118. On the Harmonic Relation of two Lines or two Points . . 96 

Phil. Mag. t. VI. (1853), pp. 105—107 

119. On a Theorem for the Development of a Factorial . . . 98 

Phil. Mag. t. VI. (1853), pp. 182—185 

120. Note on a Generalisation of the Binomial Theorem . . 101 

Phil. Mag. t VI. (1853), p. 185 

121. Note on a Question in the Theory of Prohahilities . . . IDS 

Phil. Mag. t VI. (1853), p. 259 

122. On the nomographic Transformation of a Surface of tJie Second 

Order into Itself ....... . . 10 ^^^ 

Phil. Mag. t'vi. (1853), pp. 326—333 

123. On the Geometrical Representation of an Abelian Integral . 11 — 

PhiL Mag. t vi. (1853), pp. 414—418 

124. On a Property of the Caustic by Refraction of the Circle . 1 

Phil. Mag. t VL (1853), pp. 427—431 



CONTENTS. 



IX 



125. On the Theory of Groups as depending on the Symbolical 

. Equation ^=1 

Phil. Mag. t. VII. (1854), pp. 40—47 

126. On the theory of Groups as depending on the Symbolical 

Equation ^ = 1. Second Part 

Phil. Mag. t VII. (1854), pp 408—409 

127. On the nomographic Transformation of a Surface of the 

Second Order into itself 

PhU. Mag. t VII. (1854), pp. 208—212: continuation of 122 

128. Developments on the Porism of the In-and-drcumsciibed Polygon 

PhiL Mag. t. vii. (1854), pp. 339—345 

129. On the Porism of the In-and-circuToscribed Triangle^ and on 

an irrational Transformation of two Ternary Quadratic 
Forms each into itself. ....... 

Phil. Mag. t. IX. (1855), pp. 513—517 

130. Deuxihne MSmoire sur les Fonctions doublement PSriodiques . 

Liouville, t. xix. (1854), pp. 193—208 : sequel to 25 

131. Nouvelles Recherches sur les Covariants 

CreUe, t xlvii. (1854), pp. 109—125 

132. RSponse d, une Question proposSe par M. Steiner 

Crelle, t l. (1855), pp. 277—278 

133. Sur un Thiorhme de M. Schlafli ...... 

Crelle, t. l. (1855), pp. 278—282 

134. Remarques sur la Notation des Fonctions Algebriques 

Crelle, t. l. (1855), pp. 282—285 

135. Note sur les Covariants d'une Fanction Quadratique^ OubiquCy 

ou Biquadratique d, deux IndSterminSes .... 

Crelle, t l. (1855), pp. 285—287 

136. Sur la Transformation d'une Fonction Quadratique en elle- 

mime par des Substitutions Ihieaires ..... 

Crelle, t l. (1855), pp. 288—289 

137. Seeker ches Ult&i^ieures sur les Determinants gauches 

Crelle, t. l. (1855), pp. 299—313: continuation of 52 and 69. 



PAOB 



123 



131 



133 



138 



145 



150 



164 



179 



181 



185 



189 



192 



202 




CONTENTS. 



PAOB 



138. Recherches sur les Matrices dont hs termes sont des fonctions 

linSaires d^une seule Ind4tei^min4e . . . . . 216 

Crelle, t. l. (1855), pp. 313—317 

139. An Introductory Memoir on Quantics 221 

Phil. Trans, t cxLiv. (for 1864), pp. 244—258 

140. Researches on the Partition of Numbers 235 

Phil. Trans, t. cxlv. (for 1855), pp. 127—140 

141. A Second Memoir on Quantics 250 

Phil. Trans, t. cxlvi. (for 1856), pp. 101—126 

142. Numerical Tables Supplementary to Second Memoir on Quantics 276 

Now first published (1889) 

143. Tables of the Covariants M to W of the Binary Quintic : from 

the Second, Third, Fifth, Eighth, Ninth and Tenth Memoirs 

on Quantics 282 

Arranged in the present form, 1889 

144. A Third Memmr on Quantics 310 

PhiL Trans, t. cxlvi. (for 1856), pp. 627—647 

145. A Memoir on Caustics 336 

Phil. Trans, t. cxlvii. (for 1857), pp. 273—312 

146. A Memoir on Curves of the Third Order .... 381 

Phil. Trans, t cxlvii. (for 1857), pp. 415—446 

147. A Memoir on the Symmetric Functions of the Roots of an 

Equation 417 

Phil. Trans, t. cxlvii. (for 1857), pp. 489 — 496 

148. A Memxdr on the Resultant of a System of two Equations . 440 

Phil. Trans, t. cxlvii. (for 1857), pp. 703—715 

149. On the Symmetric Functions of the Roots of certain Systems 

of two Equations ......... 454 

Phil. Trana t. cxlvii. (for 1857), pp. 717—726 

150. A Memoir on the Conditions for the Existence of given Systems 

of Equalities among the Roots of an Equation . . . 465 

Phil. Trans, t. cxlvil (for 1857), pp. 727—731 



CONTENTS. XI 



PAOX 



51. Tables of the Sturmian Functions for Equations of the Second^ 

Thirds Fourth^ and Fifth Degrees 471 

Phil. Trans, t cxLVii. (for 1857), pp. 733—736 

52. A Memoir on the Theory of MatHces 475 

Phil. Trans, t. cxLViii. (for 1858), pp. 17—37 

)3. A Memoir on the Automorphic Linear Transfoi^mation of a 

Bipartite Quadric Function 497 

Phil. Trans, t. cxlviii. (for 1858), pp. 39—46 

)4. Supplementary Researches on tJie Partition of Numbers . . 506 

PhiL Trans, t. cxlviii. (for 1858), pp. 47—52 

)5. A Fourth Memoir on Qualities 513 

Phil. Trans, t. cxlviii. (for 1858), pp. 415—427 

)6. A Fifth Memoir on Qiiantics . . . 527 

Phil. Trans, t. cxlviil (for 1858), pp. 429—460 

)7. On the Tangential of a Cubic 558 

PhiL Trans, t. cxlviil (for 1858), pp. 461—463 

8. A Sixth Memoir on Quantics 561 

Phil. Trans, t cxLix. (for 1859), pp. 61—90 



and References .......... 593 



CLASSIFICATION. 

Geometry 

Theory of Distance, 158 

Surfaces, 106, 107 

Transformation of Qnadric Surfaces, 122, 127, 129, 136, 153 



Steiner's extension of Mal&tti's Problem, 114 
In-and-circumscribed triangle and polygon, 115, 116, 128, 129 
Harmonic relation of two lines or points, 118 
Question proposed by Steiner, 132 
Caustics, 124, 145 
Cubic Curves, 146, 157 

Analysis 

Skew Determinants, 137 

Attractions and Multiple Integrals, 108 

Definite Integrals, 103 

Elliptic and Abelian Integrals, 110, 113, 117, 123, 130 

Covariants, Quantics <fec., 131, 134, 135, 139, 141, 142, 143, 144, 155, 156, 158 

Matrices, 138, 152 

Partition of Numbers, 140, 154 

Symmetric Functions dec., 147, 148, 149, 150 



Lagrange's Theorem, 101 

Double Infinite Series, 102 

Permutants, 104, 105 

Rationalisation of Algebraic Expression, 109 

Transformation of Trigonometrical Expression, 110 

Theorem of Lejeune-Dirichlet's, 111 

Products of Sums of Squares, 112 

Factorials, 119 

Generalisation of Binomial Theorem, 120 

Question in Probabilities, 121 

Groups, 125, 126 

Theorem of Schlafli's, on Elimination, 133 

Sturmian Functions, 151 



101] 



101. 



NOTES ON LAGRANGE'S THEOREM. 

[From the Cambridge and Dublin Mathematical Journal, vol. vi. (1851), pp. 37 — 45.] 

I. 

If in the ordinary form of Lagrange's theorem we write {x + a) for x, it becomes 

X = hf(a + x), 

F(a + x)=^Fa+jF'afa + &c (1) 

It follows that the equation 



F{a + x) = Fa + \j~^^{F'a/a)-i- (2) 



must reduce itself to an identity when the two sides are expanded in powers of x\ 

A 
da 

(3) 



or writing for shortness F, f instead of Fa, fa, and S for -r- , we must have 



(where p extends from to r). Or what comes to the same, 

[r]--^^°M p[P->?-[r-p]-^[>-ir' ^-^-^-^"-^^} ^*> 

where s extends from to (^ — p). The terms on the two sides which involve Z^F 
are immediately seen to be equal ; the coefficients of the remaining terms S'jP on tht^ 
second side must vanish, or we must have 



o. II, 



1 



s.^ 



2 NOTES ON Lagrange's theorem. [101 

(s being less than r). Or in a somewhat more convenient form, writing p, q and it- 
for p — 8, r—p and r — «, 

where 8 is constant and p and q vary subject to p+q = ky k being a given constant 
diflferent from zero (in the case where A:=0, the series reduces itself to the single 

term - ). The direct proof of this theorem will be given presently. 

II. 

The following symbolical form of Lagrange's theorem was given by me in the 
Mathematical Journal, vol. ill. [1843], pp. 283—286, [8]. 

If x^a^-hfx, (7) 

then 

Suppose /r = ^ (6 + h^sc\ or a? = a + A^ (6 + ky^x), then 



Fx= (j-Y'^'^F'a c*«<*+*^». 



But 

(In fact the two general terms 

{<t>{b + kita)}'^ and (^)*^e*^(<^)"', 

of which the former reduces itself to e db(if)by^, are equal on account of the equiva- 
lence of the symbols 

e**"^ and (A)*^e*^). 

Hence 

X = a •\- h4> (b •\- kyfrx), (8) 



^-=(i)*''"(^)'"^'«^*"'"^ 



and the coefficient of h^k^ is 



w 



r^(^r^'«<^'^)"(s)"('^>"- 



101] NOTES ON Lagrange's theorem. 3 

A similar formula evidently applies to the case of any finite number of functions 
0, y^, &c. : in the case of an infinite number we have 

or the coefficient of h^kH^ ... is 

mrmrY ■ :. (r«)" '" ■ {mT <♦')- ■ (s)' <+'>-• ■ • 

the last of the series m, n, p being always zero; e.g. in the coefiicient of h^k^, 

account must be had of the factor f-i-j ('^V* or (-^c)**. The above form is readily 

pn)ved independently by Taylor s theorem, without the assistance of Lagrange's. If in it 
we write h = k, &c., a = 6 = &c., and ^ = '^ = &c. =/, we have -F (a + hf{a + hf{a +...) = Fx, 
where ar = a + hfx. Hence, comparing the coefficient of A* with that given by Lagrange's 
theorem, 

whtre m + w+&c. = «, and as before Fa, /(/, .- have been replaced by F^ / 8. By 
comparing the coefficients of h^F, 

where n ■¥p-\-... =<, the last of the series n, p ,., always vanishing. The formula (10) 
deduced, as above mentioned, from Taylors theorem, and the subsequent formula (11) 
with an independent demonstration of it, not I believe materially different from that 
which will presently be given, are to be found in a memoir by M. Collins (volume ii. 
(1H38) of the Memoirs of the Academy of St Petersburg), who appears to have made 
very extensive researches in the theory of developments as connected with the combina- 
torial analysis. 

in. 

To demonstrate the formula (6), consider, in the first place, the expression 



s^^^,[m''^')m-^-^)\< 



ii^here j9 + 5 = it. Since 



1_ _1 / 1 1_ __\ 



1—2 



4 NOTES ON Lagrange's theorem. [101 

this is immediately transformed into 

= J 5 ^-j?j^j, {^ (p + 1) (p + » + !)(«>' .y>*'8/)(8«/-'--»-.) 

iu which last expression p + 9 = (^ — 1 ). Of this, after separating the factor Sf, the 
general term is 



[/>?[?-«]*- *p <^ + * + ^ <^-^> (^^z-^^-) 



equivalent to 



I f^ «' V- « f^5f {* (p + a + 1) (p + * + a + 1) (SV^+'+-) (S^/-^— -^0 

- *p (p + « + ^ (s^y*^*) W'"'''^')]* 

in which last expression p + g=A;-a — 1. By repeating the reduction j times, the 
general term becomes 

1 1 



A;(ifc-a-l)(ifc-a-/8-2) ... [a]*|j8]^... 



s-+»/.y +'/... 



x[p + » + ^ + a + /S...+j-ip-^ (SPfP+'-^-^fi") {B^f-P • » i - ^ •)}, 

where the sums a + /3... contain / terms,/ being less than j or equal to it, and S 
extends to all combinations of the quantities a, /3... taken / and / together (so that 
the summation contains 2^ terms). Also p4-3 = A; — a — )8... (J terms) — j, and the 
products A; (A; - a - 1 ) (Jfc - o - )8 - 2) . . . and [a]* [fif, . . . S*+y . ^ +y. . . contain each of 
them j terms. Suppose the reduction continued until A: — a — )8 . . . ( j terms) — j = 0, then 
the only values of p, q are p = 0, ? = ; and the general term of 



becomes 



1 1 8"+»/" ifi+^f t'-i-* 

kik-a-l){k-a-^-2)... [aj« [/SJ". . . •'•* •'-•^ 




I] NOTES ON Lagrange's theorem. 

If ^ =s 0, the general term reduces itself to 
mee finally, if ^ = — - , the general term of 



omes 



1 it is readily shown that the sum contained in this formula vanishes, which proves 
equation in question. 

IV. 

The demonstration of the equation (11) is much simpler. We have 



t is, 






"e n extends from n = 1 to n = ^. Similarly 

&C. 

ence, substituting successively, and putting t — n—p — q = r, &c., 

d the last of these corresponding to a zero value of the last of the quantities 
. is evidently the required equation (11). 

V. 

formula (18) in my paper on Lagrange's theorem (before referred to) is incorrect. 

at present, after giving the proper form of the formula in question, to 

the result of the substitution indicated at the conclusion of the paper. It 

onvenient to call to mind the general theorem, that when any number 



6 NOTES ON LAGRANQJi's THEOREM. [lOl 

of variables a?, y, z .,. are connected with as many other variables t/, r, ti; . . . by 
the same number of equations (so that the variables of each set may be considered 
as functions of those of the other set) the quotient of the expressions dxdy ... and 
dudv ... is equal to the quotient of two determinants formed with the functions which 
equated to zero express the relations between the two sets of variables ; the former 
with the differential coefficients of these functions with respect to a, » . . . , the latter 
with the differential coefficients with respect to x, y Consequently the notation 

^-T may be considered as representing the quotient of these determinants. This 

being premised, if we write 

X'-u — h0(x, y . . .) = 0, 

y-V'-k<f>(x, y ...) = 0, 
then the formula in question is 

if for shortness the letters 0, 0, . . . , F denote what the corresponding functions become 

when M, v, ... are substituted for x, y, — Let -r denote the value which , j '" , 

A dudv... 

considered as a function of x, y . . . , assumes when these variables are changed into 
w, V, . . . , we have 



V = 



1-AM, -AS^d... !. 

— KOu^» 1 — A?Ov0 . . . 



By changing the function F, we obtain 

Fix, y...) = S«***S^*^*...«*^**- .FV; 

where, however, it must be remembered that the A, A:, ... , in so far as they enter into 

the function V, are not aflfected by the symbols AS^, AS*,... In order that we may 

consider them to be so affected, it is necessary in the function V to replace A, A:, &c. 

h k 
by ^ , ^ , &c. Also, afler this is done, observing that the symbols ASu^, hB„0 ... affect 

Ou Off 

a function gW+u-h ... /^ xh^ symbols hBuO, hB„0,.,. may be replaced by S,/, S/, ..., where 
the is not an index, but an affix denoting that the differentiation is only to be 
performed with respect to u, t; ... so far as these variables respectively enter into 
the function 0. Transforming the other lines of the determinant in the same manner, 

and taking out from Su "K * ••• the factor SuB^ ... in order to multiply this last 
factor into the determinant, we obtain 

Fij-, y...) = S„"»- S."'-' . . . ««-*♦• F D ; 
where 

n= ««-««*. -««♦.... , 




101] NOTES ON Lagrange's theorem. 7 

in which expression S„, S„... are to be replaced by 

The complete expansion is easily arrived at by induction, and the form is somewhat 
singular. In the case of a single variable u we have □ = Su, in the case of two 
variables, □ = S„'S„' + S„'S„^ + K^K^- Or writing down only the aflSxes, in the case of 
a single variable we have F] in the case of two variables FF, Fd, <f)F] and in the 
case of three variables FFF, <f>FF, x^^y ^X^> ^^^> ^^^. ^^<^» ^^^> ^H* ^X^» 0^0. 
X-F<^, 4>F0, xxF^ ^X^> X^^'i where it will be observed that 6 never occurs in the 
first place, nor ^ in the second place, nor d, <f> (in any order) in the first and second 
places, &c., nor 0, (f>, x 0^ *^y order) in the first, second, and third places. And the 
same property holds in the general case for each letter and binary, ternary, &c. 
combination, and for the entire system of letters, and the system of affixes contains 
every possible combination of letters not excluded by the rule just given. Thus in the 
case of two letters, forming the system of aflBxes FF, F0, tf>F, 0F, F<\>, 0<f>, <f)0, the last 
four are excluded, the first three of them by containing in the first place or <f> 
in the second place, the last by containing <l>, in the first and second places : and 
there remains only the terms FF, F0, ^F forming the system given above. Substituting 
the expanded value of □ in the expression for F (a;, y...), the equation may either be 
permitted to remain in the form which it thus assumes, or we may, in order to 
obtain the finally reduced form, after expanding the powers of A, A: . . . , connect the 
symbols S^*, S„*...Su', &c. with the corresponding functions 0, <f).,.F, and then omit the 
affixes ; thus, in particular, in the case of a single variable the general term of Fx is 

(the ordinary form of Lagranges theorem). In the case of two letters the general 
term of F{xy y) is 

(see the MScaniqvs Celeste, [Ed. 1, 1798] t. i. p. 176). In the case of three variables, 
the general term is 

MWW' ^'^'^''"*"''"' {^*V«AS^ +...}. 

the sixteen terms within the { } being found by comparing the product S„SpS«, with 
the system FFF, 4>FF^ &c., given above, and then connecting each symbol of diff'eren- 
tiation with the function corresponding to the aflfix. Thus in the first term the 
^> ^vt ^vi ^a^b affect the F, in the second term the h^ affects ^^, and the 8^ and h^ 
each affect the F, and so on for the remaining terms. The form is of course deducible 
from Laplace's general theorem, and the actual development of it is given in Laplace's 
Memoir in the Hist, de VAcad. 1777. I quote from a memoir by Jacobi which I take 
this opportunity of referring to, "De resolutione equationum per series infinitas," 
CreUe, t. vi. [1830], pp. 257 — 286, founded on a preceding memoir, "Exercitatio algebraica 
circa discerptionem singularem fractionum quae plures variabiles involvunt," t. V. [1830], 
pp. 344—364. 

Stone Buildings, April 6, 1850. 



8 



[102 



102. 



ON A DOUBLE INFINITE SERIES. 



[Fn)m the Cambridge and Dublin Mathematical Journal, vol. vi. (1851), pp. 45 — 47.] 

The following completely paradoxical investigation of the properties of the fianction 
r (which I have been in possession of for some years) may perhaps be found interesting 
from its connexion with the theories of expansion and divergent serieis. 

Let Sr^r denote the sum of the values of ifyr for all integer values of r from 
— X to X . Then writing 

w = 2^[n-l]'-a:«-»-^, (1) 

(where n is any number whatever), we have immediately 

^'- = 2r [n - ly-^^ af-*^ ^-^rin- If x^"'-'' = u ; 

1 . du ^ 

that IS, ^ ~ ^» ^' ^ ^ ^»»^» 

(the constant of integration being of course in general a function of n). Hence 

(7n^ = 2r[n-l]'*ic~-^-^; (2) 

or 6* is expanded in general in a doubly infinite necessarily divergent series of /fractional 

powers of x, (which resolves itself however in the case of n a positive or negative 

integer, into the ordinary singly infinite series, the value of Cn in this case being 
immediately seen to be Fn). 

The equation (2) in its general form is to be considered as a definition of the 
function (?». We deduce from it 

Xr [n - 1]'- (flur)'»->-^ = C„e«* , 
1^ [n' - ly {aaff-^-^' = Cn'^ ; 



102] ON A DOUBLE INFINITE SERIES. 9 

and also 

2jk [n + n' . . . - 1]* {a (a? + a?' . . . )}«+n'...-i-* = (?«+«•... c«<«+-^ •>. 

Multiplying the first set of series, and comparing with this last, 

Cn+n' ...2.. ^ ... [n - l]*- [n' ^ly... a;~-^- a/«-»-^ . . . 

= CnCn'...[n + n'...-l]*(a; + a;'. ..)•'"'**'•""'"*, (3) 

(where r, r denote any positive or negative integer numbers satisfying r + r'+...=A:+l— p, 
p being the number of terms in the series n, n\...). This equation constitutes a 
multinomial theorem of a class analogous to that of the exponential theorem contained 
in the equation (2). 

In particular 

C^n' ... 2,y ... [n-iy [n' - ly . . . = CnC^. . . . [n + n' . . . - l]*|,»+«'...-i-t, (4) 

and if p = 2, writing also m, n for n, n', and k—l — r for r*, 

C„+,2,[m-l]'[«-l]*-'-'=0„C„[m + n-l]*2»+"-'-* (5) 

or putting k = and dividing, 

C„(7„-C„+„ = 2„^-2,[m-l]'[n-l]-'-'. (6) 

Now the series on the second side of this equation is easily seen to be convergent 
(at least for "positive values of m, n). To determine its value write 





then 



(m, n) = I af^^ (1 — w)^^ dx ; 

J 



F(m, n)= I a?^»(l~ir)«-*(ir+ f af"-^ {I - xf^^ dx \ 

J Q J Q 

and by successive integrations by parts, the first of these integrals is reducible to 

1 

^_^-^Y 2r [w — l]*" [^ — l]"^"', ^ extending from — 1 to — x inclusively, and the second to 

^^li^^^zi ^r [^ — l]*" [w — l]""*~^i ^* extending from to oo ; hence 

or C„C„-.C^„ = F(m, n) (7) 

C. 11. 2 



10 ON A DOUBLE INFINITE 8ERI1*>$. [102 

which proves the identity of Cn^ with the function T (m), {Substituting in two of the 
preceding equations, we have 

TnTn'. . . - T (n + r' . . .) = [„+„'... _l]fcyn-^n..-.-t ^ry... [n - 1]' [n' - 1]' (8) 

(where, as before, p denotes the number of terms in the series n, n\... and r+r'4-...=ifc+l— /)), 
the first side of which equation is, it is well known, reducible t-o a multiple definite 
integral by means of a theorem of M. Dirichlet's. And 

''("'• ") = [,« + n-iy 2"^»— * ^' ^"' - ^^' t» - 1]*— ^ (9) , 

where r extends from — x to 4- « , and k is arbitrary. By giving large negative 
values to this quantity, very convergent series may be obtained for the calculation of 
F(m, n)]. 




103] 



11 



103. 



ON CERTAIN DEFINITE INTEGRALS. 

[From the Cambridge and Dvblin Mathematical Journal, vol. vi. (1851), pp. 136 — 140.] 

Suppose that for any positive or negative integral value of r, we have -^(rc + ra) 
= Ur yp'Xy Ur being in general a function of x, and consider the definite integral 



J — 00 



'a being any other function of x. In case of either of the functions yp'x, 'Vx becoming 
iTiiinite for any real value a of x, the principal value of the integral is to be taken, 
that is, 7 is to be considered as the limit of 



(j + f* '^irx^xdx, (€ = 0), 



a»n.d similarly, when one of the functions becomes infinite for several of such values 
We have 

/ r(r+i)a \ 

/ = (... I +...j^a?^«da?; 

^^ changing the variables in the different integrals so as to make the limits of each 
^» O, we have 

1=1 [2'^(ic + ra)^(a; + ra)]da?, 

Jo 

"^ Extending to all positive or negative integer values of r, that is, 

I=ryftx[XUr'ir{x + ra)]dx, (A) 

Jo 

2—2 



i 



12 



ON CERTAIN DEFINITE INTEGRALS. 



[103 



which is true, even when the quantity under the integral sign becomes infinite for 
particular values of x, provided the integral be replaced by its principal value, that is, 
provided it be considered as the limit of 



or 






where a, or one of the limiting values a, 0, is the value of x, for which the quantity 
under the integral sign becomes infinite^ and 6 is ultimately evanescent. 

In particular, taking for simplicity a = tt, suppose 

-^ (a? 4- tt) = ± -^x, or -^ (a? + nr) = {±y y^x ; 



then observing the equation 



Z — - — = cot X, or = cosec x, 
x + nr 



according as the upper or under sign is taken, and assuming 'Vx = x~*^, we have finally 

the former equation corresponding to the case of -^ (j? + tt) = y^x^ the latter to that of 
'^ (ic 4- tt) = — y^x. 

Suppose y^^x = yp'gx, g being a positive integer. Then 

r * yjr^xdx _ ^ f * y^xdx 

also if >fr (j; + -w) = i^jr, then -^^ (;c + tt) = >^,a: ; but if -^/r (.-c + tt) = — i^J-, then '^^{x-^-ir) 
= ±ylr^x, the upper or under sign according as g is even or odd. Combining these 
equations, we have 

y^(x-{-7r) = yp'X, g even or odd, 
•^ (a; 4- tt) = — '^a?, g even. 



' cot x] At = < j,^ - j' ^x [(J^J" ' cot x] .ir ; 



/.: 



fgxdx _(^y-^ f 



'^ (a? 4- "w) = — -^a?, g odd, 



g^-'^\x 



-T- 1 cosec 






/-.^=S>-V-/>-« |-0-«^^^^^-| ^^(..^'IJ^x [(^"'cosecx] d,. 




103] ON CERTAIN DEFINITE INTEGRALS. 13 

In particular 



sinxdx 



f Sin a 

J -00 *^ 

/ sin ^07 f -T J cot j: cir = (/**"* / sina: (j-j coseca? ch, g even, 

I sin^ra: f-j-j coseeo: c2a?= 9^"^] '^^^^ (;/") coseca: cir, jr odd, 

I sin gxcotxdx^Tr, g even, 

I sin ^a: cosec iTcia? = TT, g odd, 

C'tSLUxdx ^ « 

= 0, i&c., 

the number of which might be indefinitely extended. 

The same principle applies to multiple integrals of any order: thus for double 
integrals, if '^(x + ra, y + rb) = Ur ,i'^ (a:, y), then 

I i ir{x, y)^(x, y)dxdy^r ( fix, y) 2 £7,,, ^ (a? + ra, y + sb). ... (B) 

J— 3oJ-ao J J 

In particular, writing WyV for a, 6, and assuming y^(x + rWf y + sv) = (±y {±y yfr (x, y); 
also '^(x, y) = (x + iy)~'^, where as usual i=\/ — 1» 



where 



-a/ . • N x' (±X(±)'l 
^ ^^ (x + ty -{-rw + sm) 



S extending to all positive or negative integer values of r and s. Employing the 
notation of a paper in the Cambridge Mathematical Journal, "On the Inverse Elliptic 
Functions," t. iv. [1845], pp. 257 — 277, [24], we have for the diflferent combinations 
of the ambiguous sign, 

^/ • V iS(x+iy) 1 

1. -, -, e(a? + ty)= , 7'\ =j./-r'-\> 

^ ^^ y{x + %y) ^(a? + ty) 



14 ON CERTAIN DEFINITE INTEGRALS. [103 



3. + 



ft / 1 • \ _ g (a? 4- iy ) _ /(a? - f iy) 



4. +, +, e 



/ . • \ 7'(^+*y) 



where ^, /, ^ are in fact the symbols of the inverse elliptic functions (Abel's notation) 
corresponding very nearly to sin am, cos am, A am. It is remarkable that the last 
value of cannot be thus expressed, but only by means of the more complicated 
transcendant yx, corresponding to the H(x) of M. Jacobi. The four cases correspond 
obviously to 

1 . 1^ (a? + rw, y + w) = (-)''+• yjr (», y), 

2. -^(x-^-rw, y + 8v) = (-y ^{x, y), 

3. ^{x-^-rw, y + 8v) = (-'y y^(x, y\ 

4. y}r(x + rw, y + 8v)= y^(x, y). 

The above formulae may be all of them modified, as in the case of single integrals, 
by means of the obvious equation 

The most important particular case is 



00 # 00 



^ 00 # 

/.J- 



00 



(* + iy) 



for in almost all the others, for example in 

the second integration cannot be effected. 

Suppose next -^{x, y) is one of the functions 7(a? + iy), g{x + iy), G(x-\'iy\ 
CS (a? + iy), so that 

^|r(x-\'rw, y + 8v) = (±y{±yUrjylr{x, y\ 
where 

(see memoir quoted). Then, retaining the same value as before of "9 (x, y), we have 
still the formula (B), in which 



^ ^^ X + ly '\-rw + 8m 



But this summation has not yet been effected; the difficulty consists in the variable 
factor €^* ('•«'-«*') in the numerator, nothing being known I believe of the decomposition 
of functions into series of this form. 



103] 



ON CERTAIN DEFINITE INTEGRALS. 



15 



On the subject of the preceding" paper may be consulted the following memoirs by 
Raabe, "Ueber die Summation periodischer Reihen," Crelle, t. xv. [1836], pp. 355 — 364, 
and •* Ueber die Summation harmonisch periodischer Reihen," t. xxiil. [1842], pp. 105 — 
125, and t. xxv. [1843], pp. 160 — 168. The integrals he considers, are taken between 
the limits 0, oo (instead of — oo , oo ). His results are consequently more general than 
those given above, but they might be obtained by an analogous method, instead of 
the much more complicated one adopted by him : thus if <^ (a? -h 2'rr) = <f)x, the integral 



/ 



90 



if>x — reduces itself to 



^•r*'?Ssr-r'^*'[i+^"(iA™-2iy • 



provided I dx(t>x = 0. The summation in this formula may be effected by means of 

Jo 

the function F and its differential coefficient, and we have 



/: 



, dx 

^ X 






^'^) 



which is in effect Raabe's formula (10), Crelle, t. xxv. p. 166. 

By dividing the integral on the right-hand side of the equation into two others 
whose limits are 0, tt, and tt, 27r respectively, and writing in the second of these 27r — a: 
instead of x, then 



J ^ 29r j J 



<l>x i:^-h<^(27r-a:)— ^ 



27r>' 



^ 



'•'£) 



f '>-£)' 



dx; 



or reducing by 



■- (4) '■'('- s) 



— TT cot ^X, 



we have 



--^^ 



j ^ — =ij <l>xcoHxdx-^j [4>x -h (f> (in - x)] — JT^» 

^"ich corresponds to Raabe's formula (10'). If <^ (- aj) = - <f>x, so that ^ -h <^ (27r — a?) = 0, 

^^^ last formula is simplified ; but then the integral on the first side may be replaced 

k f * dx 

"Ml <l>x — ,80 that this belongs to the preceding class of formulse. 

</ — ae X 



16 



[104 



104. 



ON THE THEORY OF PERMUTANTS. 



[From the Cavibridge and Dublin Maihematical Journal, vol. vii. (1852), pp. 40 — 5l.] 

A FORM may by considered as composed of blanks which are to be filled up by 
inserting in them specializing characters, and a form the blanks of which are so filled 
up becomes a symbol. We may for brevity speak of the blanks of a symbol in the 
sense of the blanks of the form from which such symbol is derived. Suppose the 
characters are 1, 2, 3, 4,..., the symbol may always be represented in the first 
instance and without reference to the nature of the form, by F1334... And it will be 
proper to consider the blanks as having an invariable order to which reference will 
implicitly be made; thus, in speaking of the characters 2, 1, 3, 4,... instead of as 
before 1, 2, 4,... the symbol will be V^^,., instead of V^^,.. , When the form is 
given we shall have an equation such as 

according to the particular nature of the form. 

Consider now the characters 1, 2, 3, 4,..., and let the primitive arrangement and 
every arrangement derivable from it by means of an even number of inversions or 
interchanges of two characters be considered as positive, and the arrangements derived 
from the primitive arrangement by an odd number of inversions or interchanges of 
two characters be considered as negative ; a rule which may be termed " the rule of 
signs." The aggregate of the symbols which correspond to every possible arrangement 
of the characters, giving to each symbol the sign of the arrangement, may be termed 
a ** Permutant ; " or, in distinction from the more general functions which will presently 
be considered, a simple permutant, and may be represented by enclosing the sjTnbol 
in brackets, thus {V^^,,,). And by using an expression still more elliptical than the 
blanks of a symbol, we may speak of the blanks of a permutant, or the characters 
of a permutant. 



104] ON THE THEORY OP PERMUTANT8. 17 

As an instance of a simple permutant, we may take 

(r^)^r^+ v^ + r^,^ f«- f«- f„; 

and if in particular Fia = ai6j(J», then 

It follows at once that a simple permutant remains unaltered, to the sign prhs according 
to the rule of signs, by any permutations of the characters entering into the per- 
mutant For instance, 

(F-„) = (F„) = (F„) = - ( F«.) = - (F„) = - (F„). 

Consequently also when two or more of the characters are identical, the permutant 
vanishes, thus 

The form of the symbol may be such that the sjrmbol remains unaltered, to the sign 
pris according to the rule of signs, for any permutations of the characters in certain 
particular bUmka Such a system of blanks may be termed a quote. Thus, if the first 
and second blanks are a quote, 

and consequently 

(F„) = 2(F„+F„+F„); 

and if the blanks constitute one single quote, 

( ^m . . . ) = iV K ug . . . , 

where iV=1.2.3...w, n being the number of characters. An important case, which 
will be noticed in the sequel, is that in which the whole series of blanks divide 
themselves into quotes, each of them containing the same number of blanks. Thus, 
if the first and second blanks, and the third and fourth blanks, form quotes respectively, 

It is easy now to pass to the general definition of a "Permutant." We have only 
to consider the blanks as forming, not as heretofore a single set, but any number of 
distinct sets, and to consider the characters in each set of blanks as permutable 
inter ae and not otherwise, giving to the sjnnbol the sign compounded of the signs 
corresponding to the arraDgements of the characters in the different sets of blanks. 
Thus, if the first and second blanks form a set, and the third and fourth blanks form 
a set, 

The word 'set' will be used throughout in the above technical sense. The particular 
mode in which the blanks are divided into sets may be indicated either in words or 
by some superadded notation. It is clear that the theory of permutants depends 
ultimately on that of simple permutants; for if in a compound permutant we first 
write down all the terms which can be obtained, leaving unpermuted the characters 
c. II. 3 



18 ON THE THEORY OP PERMUTANT8. [104 

of a particular set, and replace each of the terms so obtained by a simple permutant 
having for its characters the characters of the previously unpermuted set, the result 
is obviously the original compound permutant. Thus, in the above-mentioned case, 
where the first and second blanks form a set and the third and fourth blanks form 
a set 

((f;»o)-=(^i~)-(f.^). 

in the former of which equations the first and second blanks in each of the permutants 
on the second side form a set, and in the latter the third and fourth blanks in each 
of the permutants on the second side form a set, the remaining blanks being simply 
supernumerary and the characters in them unpermutable. It should be noted that 
the term quote, as previously defined, is only applicable to a system of blanks belonging 
to the same set, and it does not appear that anything would be gained by removing 
this restriction. 

The following rule for the expansion of a simple permutant (and which may be 
at once extended to compound permutants) is obvious. Write down all the distinct 
terms that can be obtained, on the supposition that the blanks group themselves in 
any manner into quotes, and replace each of the terms so obtained by a compound 
permutant having for a distinct set the blanks of each assumed quote; the result is 
the original simple permutant. Thus in the simple permutant (Vmd, supposing for 
the moment that the first and second blanks form a quote, and that the third and 
fourth blanks form a quote, this leads to the equation 

( F;«) = + (( F,^)) + (( r^)) + (( F,«)) + (( F^)) + (( F„,)) + (( F«„)), 

where in each of the permutants on the second side the first and second blanks form 
a set, and also the third and fourth blanks. 

The blanks of a simple or compound permutant may of couree, without either 
gain or loss of generality, be considered as having any particular arrangement in qaaoe, 
for instance, in the form of a rectangle : thus F„ is neither more nor less general than 

Fis4. The idea of some such arrangement naturally presents itself as affording a means 
of showing in what manner the blanks are grouped into seta But, considering the 
blanks as so arranged in a rectangular form, or in lines and columns, suppose in the 
first instance that this arrangement is independent of the grouping of the blanks into 
sets, or that the blanks of each set or of any of them are distributed at random in 
the different lines and columns. Assume that the form is such that a sjnnbol 

'^«^y ... 

is a function of symbols Vafiy..., Va^y'...t &c. Or, passing over this general case, and 
the case (of intermediate generality) of the function being a symmetrical 'function, 
assume that 

afi^y,.. 



104] ON THE THEORY OF PBRMUTANXa 19 

is the product of symbols V^i^y..., V.js'y..., &c Upon this assumption it becomes 
important to distinguish the different ways in which the blanks of a set are distributed 
in the different lines and columns. The cases to be considered are : (A). The blanks 
of a single set or of single sets are situated in more than one column. (E), The 
blanks of each single set are situated in the same column. (C). The blanks of each 
single set form a separate column. The case (B) (which includes the case (C)) and the 
case (C) merit particular consideration. In fact the case (E) is that of the functions 
which I have, in my memoir on Linear Transformations in the Journal, [13, 14] 
called hyperdeterminants, and the case ((7) is that of the particular class of hyper- 
determinants previously treated of by me in the Cambridge PhUoaophical Tranaactiona, 
[12] and also particularly noticed in the memoir on Linear Transformations. The 
functions of the case (B) I now propose to call '' Intermutants," and those in the case 
(C) *' Commutants." Commutants include as a particular case '' Determinants," which 
term will be used in its ordinary signification. The case (A) I shall not at present 
discuss in its generality, but only with the further assumption that the blanks form a 
single set (this, if nothing further were added, would render the arrangement of the 
blanks into lines and columns valueless), and moreover that the blanks of each line 
form a quote: the permutants of this class (from their connexion with the researches 
of Pfaflf on differential equations) I shall term "Pfaffians." And first of commutants, 
which, as before remarked, include determinants. 

The general expression of a commutant is 

(^11 ); or ai ...^ 



11... 
22 

nn 



22 



nn 



J 



and (stating again for this particular case the general rule for the formation of a 
permutant) if, permuting the characters in the same column in every possible way, 
considering these permutations as positive or negative according to the rule of signs, 
one system be represented by 

'1 " 1 • • • 

the commutant is the sum of all the different terms 

The different permutations may be formed as follows: first permute the characters in 
all the columns except a single column, and in each of the arrangements so obtained 
permute entire lines of characters. It is obvious that, considering any one of the 
arrangements obtained by permutations of the characters in all the columns but one, 
the permutations of entire lines and the addition of the proper sign will only reproduce 

3—2 



20 



ON THE THEORY OP PERMUTANT8. 



[104 



the same 83nnbol — in the case of an even number of columns constantly with the 
positive sign, but in the case of an odd number of columns with the positive or 
negative sign, according to the rule of signs. For the inversion or interchange of two 
entire lines is equivalent to as many inversions or interchanges of two characters as 
there are characters in a line, that is, as there are columns, and consequently intro- 
duces a sign compounded of as many negative signs as there are columns. Hence 

Theoreh a commutant of an even number of columns may be calculated by 
considering the characters of any one column (no matter which) as supernumerary 
unpermutable characters, and multiplying the result by the number of permutations of 
as many things as there are lines in the commutant 

The mark -f* added to a commutant of an even number of columns will be employed 
to show that the numerical multiplier is to be omitted. The same mark placed over 
any one of the columns of the commutant will show that the characters of that 
particular column are considered as non-permutable. 

A determinant is consequently represented indiflferently by the notations 



11^ 


t 


t 


> 


+ 

ir 


22 

• 




22 

• 




22 

• 


• 




• 

, KM , 




• 



and a commutant of an odd number of cohimns vanishes identically. 

By considering, however, a commutant of an odd number of columns, having the 
characters of some one column non-permutable, we obtain what will in the sequel be 
Hpoken of as commutants of an odd number of columns. This non-permutability will be 
denoted, as before, by means of the mark f placed over the column in question, and 
it is to be noticed that it is not, as in the case of a commutant of an even number 
of columns, indifferert over which of the columns the mark in question is placed; and 
consequently there would be no meaning in simply adding the mark f to a com- 
mutant of an odd number of columns. 

A commutant is said to be symmetrical when the symbols Fo/jy... are such as to 
remain unaltered by any permutations inter se of the characters a, /8, 7 . . . A com- 
mutant is said to be skew when each symbol V^py, is such as to be altered in sign 
only according to the rule of signs for any permutations inter se of the characters 
a, /9, 7 . . . , this of course implies that the symbol Va^y.,. vanishes when any two of 
the characters a, 13, 7... are identical. The commutant is said to be demi-skew when 
Fa,^.y... is altered in sign only, according to the rule of signs for any permutation 
inter se of non-identical characters a, /8, 7,... 

An intermutant is represented by a notation similar to that of a commutant. The 
sets are to be distinguished, whenever it is possible to do so, by placing in contiguity 
the symbols of the same set, and separating them by a stroke or bar from the symbols 



104] 



ON THE THEORY OP PERMUTANT8. 



21 



of the adjacent sets. If, however, the symbols of the same set cannot be placed con- 
tiguously, we may distinguish the symbols of a set by annexing to them some auxiliary 
character by way of su£Bx or otherwise, these auxiliary symbols being omitted in the 

final result. Thus 

ri 1 la) 



2 
3 

4 



2 
3 
3 



2b 



oa 



6b 



would show that 1, 2 of the first column and the 3, 4 of the same column, the 1, 2 
and the upper 3 of the second column, and the lower 3 of the same column, the 1, 5 
of the third column, and the 2, 6 of the same column, form so many distinct sets, — 
the intermutant containing therefore 

(2.2.6.1.2.2 = ) 96 terms. 

A commutant of an even number of columns may be considered as an intermutant 
such that the characters of some one (no matter which) of its columns form each of 
them by itself a distinct set, and in like manner a commutant of an odd number of 
columns may be considered as an intermutant such that the characters of some one 
determinate column form each of them by itself a distinct set 

The distinction of sjnmmetrical, skew and demi-skew applies obviously as well to 
intermutants as to commutants. The theory of skew intermutants and skew commutants 
has a connexion with that of Pfa£Bans. 

Suppose F.^y... = V^+fi+y... (which implies the sjrmmetry of the intermutant or com- 
mutant) and write for shortness F© = a, Fi = 6, Fj = c, &c. Then 



\i 





1 







1 



T 

[j J]=(ac-n &C. 



The functions on the second side are evidently hyperdeterminants such as are 
discussed in my memoir on Linear Transformations, and there is no diflSculty in 
forming directly from the intermutant or commutant on the first side of the equation 
the symbol of derivation (in the sense of the memoir on Linear Transformations) from 
which the hyperdeterminant is obtained. Thus 



ri' 



is 12 . UU, 



r]' 



is liU-oU', 








1 



1 





1 



1 






is 12 .UU. 



12U'U\ 



22 



ON THE THEORY OF PEBMUTANTS. 



[104 



each pennutable column corresponding to a 12(') and a non-permutable column 

1 1 

changing JJV into U'U'K Similarly 



CO 0^ 
1 1 

l2 2) 



becomes (12 . 13 . 23)* . UUU, 



ro 


t 
0] 


1 


1 


.2 


2) 



becomes 12. 13.28 U'*V^^U\ 



ro 


0] 


1 


1 


2 


2 


3 

V 


sj 



zr— _— . S 



becomes (12 . IS . 14 . 23 . 24 . 84) UUUU, &c 



The analogy would be closer if in the memoir on Linear Transformations, just as 
12 is used to signify 






, 123 had been used to signify 






kic,, for 



then 



ro 


0] 


1 


1 


u 


2) 



would have corresponded to 123 .UUU, 



ro 


0] 


1 


1 


u 


2} 



to 128 £r-»[7'I7»; and this 



would not only have been an addition of some importance to the theoij*, but would 
in some instances have facilitated the calculation of hyperdeterminants. The preceding 
remarks show that the intermutant 

ro 0^ 
1 1 T 



U 1 ^J 

(where the first and fourth blanks in the last column are to be considered as belonging 
to the same set) is in the hyperdeterminant notation (12 . 34)*.(14. 23) [7Z7[7£r. 

1 Viz. corresponds to l2 beoaase and 1 are the characters ooonpying the first and second blanks of a oolomn. 
1 
If and 1 had been the characters occapying the second and third blanks in a column, the symbol would have been 

23 and so on. It will be remembered, that the symbolic nombers 1, 2 in the hyperdeterminant notation are 

merely introdaoed to distinguish from each other functions which are made identical after certain differentiations 
are performed. 



104] 



ON THE THEORY OF PERMUTANTS. 



?3 



It will, I think, illustrate the general theory to perform the development of the 
last-mentioned intermntant. We have 



'0 0^ 


as 


+ 
0' 


— 


t 

0' 


— 


t 

'0 r 


+ 


t 

^0 1^ 


111 




1 I 1 




110 




1 1 1 




1 1 












1 




1 




1 


ll 1 IJ 




a 1 1. 




.111- 




A 1 0. 




.1 1 0> 



^2 (TO on ro oi-ro o] ro o it 
iLi 1 iJ Ll 1 iJ Ll 1 oj Ll 1 iJ 

= 2{(ad-6c)«-4(ac-6»)(W-c»)}, 
= 2 (a»(? + 4ac» + 46»rf - 36»c» - 6abcd), 
the different steps of which may he easily verified. 

The following important theorem (which is, I believe, the same as a theorem of 
Mr Sylvester's, published in the Philosophical Magazine) is perhaps best exhibited by 
means of a simple example. Consider the intermutant 



'x 


r 


y 


4 


X 


3 


u 


2. 



where in the first column the sets are distinguished as before by the horizontal bar, 
but in the second column the 1, 2 are to be considered as forming a set, and the 
3, 4 as forming a second set. Then, partially expanding, the intermutant is 



'x 


r 


— 


'y 


V 


— 


'x 


V 


+ 


'y 


r 


y 


4 




X 


4 




y 


4 




X 


4 


X 


3 




X 


3 




y 


3 




y 


3 


y 


2. 




y 


2. 




<x 


2> 




.X 


2. 



or, since entire horizontal lines may obviously be permuted, 



+ 

'x V 


— 


t 
'y 


r 


•^ 


+ 
'x 


I' 


+ 


t 
> 1^ 


y 2 




y 


2 




X 


2 




X 2 


X 3 




X 


3 




y 


3 




y 3 

^X 4; 


.y 4> 




.« 


4> 




.y 


*. 





24 



ON THE THEORY OF PERMUTANTS. 



[104 



and, observing that the 1, 2 form a permutable system as do also the 3, 4, the 
second and third terms vanish, while the first and fourth terms are equivalent to 
each other; we may therefore write 



'x 


V 


s 


'x 


V 


y 


2 




y 


4 


X 


3 




X 


3 


.y 


4> 




.y 


2. 



where on the first side of the equation the bar has been introduced into the second 
column, in order to show that throughout the equation the 1, 2 and the 3, 4 are 
to be considered as forming distinct sets. 

Consider in like manner the expression 

(x U 



y 

z 



7 
6 



X 8 

y 2 

z 9 
X 4 

y 5 

^z Sj 

where in the first column the sets are distinguished by the horizontal bars and in 
the second column the characters 1, 2, 3 and 4, 5, 6 and 7, 8, 9 are to be 
considered as belonging to distinct sets. The same reasoning as in the former case 
will show that this is a multiple of 



X 


V 


y 


2 


z 


3 


X 


"i 


y 


5 


z 


6 


X 


7 


y 


8 


J 


9. 



and to find the numerical multiplier it is only necessary to inquire in how many 
wajTS, in the expression first written down, the characters of the first column can be 



104] 



ON THB THEORY OF PBRMUTANTS. 



25 



permuted so that x, y, z may go with 1, 2, 3 and with 4s, 5, 6 and with 7, 8, 9. 

The order of the x, y, z ia the second triad may be considered as arbitrary; but 

onoe assumed, it determines the place of one of the letters in the first triad; for 

instance, xS and z9 determine y7. The first triad must therefore contain xl and z6 

or x6 and zl. Suppose the former, then the third triad must contain zS, but the 

remaining two combinations may be either x4i, y5, or x5, y4. Similarly, if the first 

triad contained x6, zl, there would be two forms of the third triad, or a given 

form of the second triad gives four different forms. There are therefore in all 

24 forms, or 

t 
24 



'x V 


= 


'x r 


y 2 




y 7 


z 3 
X 4 




z 6 
X 8 


y 5 




y 2 


z 6 
X 7 




z 9 
X 4 


y 8 




y 5 


.-^ 9. 




z 3 



where the bars in the second column on the first side show that throughout the 
equation 1, 2, 3 and 4, 5, 6 and 7, 8, 9 are to be considered as forming distinct 
sets. The above proof is in reality perfectly general, and it seems hardly necessary 
to render it so in terms. 

To perceive the significance of the above equation it should be noticed that the 
first side is a product of determinants, viz. 



24 



'x 


r 


t 


'x 


6^ 


t 


'x 


7' 


y 


2 




y 


6 




y 


8 


.z 


3. 




.z 


7. 




.z 


9. 



t; 



and if the second side be partially expanded by permuting the characters of the 
second column, each of the terms so obtained is in like manner a product of deter- 
minants, so that 



24 



/ 

X 


r 


t 


'x 


4^ 


t 


'x 


T 


t = 


'x 


V 


t 


'x 


8^ 


t 


'x 


4^ 


y 


2 




y 


5 




y 


8 




y 


7 




y 


2 




y 


5 


.e 


3. 




.e 


6. 




.» 


9> 




.z 


6. 




.« 


9. 




.« 


3. 



+ ±&c., 



the permutations on the second side being the permutations inter se of 1, 2, 3, of 
4, 5, 6, and of 7, 8, 9. 

It is obvious that the preceding theorem is not confined to intermutants of two 
columns. 

c. n. 4 



26 ON THE THEORY OF PBRMUTANTS. [104 



POSTSCRIPT. 

I wish to explain as accurately as I am able, the extent of my obligations to Mr Sylvester in 
respect of the subject of the present memoir. The term permutant is due to him — ^intemiutant and 
commutant are merely terms framed between us in analogy with permutant, and the names date from 
the present year (1851). The theory of commutants is given in my memoir in the Cambridge PhUo- 
iophical Transactionij [12], and is presupposed in the memoir on Linear Transformations, [13, 14]. It 
will appear by the last-mentioned memoir that it was by representing the coefficients of a biquadratic 
function by a = 1111, 6 = 1112 = 1121 «=&c, c=1122s&c., c?=1222 = &c., 6 = 2222, and forming the 
commutant Mlll^ that I was led to the function (w-4W+3c*. The function aoe+ibcd-cuP-h-e-i"^ 

[ 2222 J 
or a, 6, c is mentioned in the memoir on Linear Transformations, as brought into notice by 

bf Cj d 

c, dy e 

Mr Boole. From the particular mode in which the coefficients a, 6,... were represented by symUdH 
such as 1111, &c., 1 did not perceive that the last-mentioned function could be expressed in the 
commutant notation. The notion of a permutant, in its most general sense, is explained by me in 
my memoir, "Sur les determinants gauches,'' Ordley t xzzvii. pp. 93 — 96, [69]; see the paragraph 
(p. 94) commencing " On obtient ces fonctions, &c,** and which should run as follows : *' On obtieiit 
oes fonctions (dont je reprends ici la thdorie) par les propriety gdnerales d'un determinant d^fini 

comme suit. En exprimant &c. ;" the sentence as printed being " d^fini. Car en exprimant &c.,'' 

which confuses the sense. [The paragraph is printed correctly 69, p. 411.] Some time in the presieut 
year (1851) Mr Sylvester, in conversation, made to me the very important remark, that as one of a 
class the above-mentioned function, 

aoe + ibcd -ad^^l^-c^, 

could be expressed in the commutant notation ( ^ , viz. by considering 00 = a, 01 «» 10 = />, 

1 1 
.2 2; 
02 = llai20=c, 12»21=cf, 22 =e; and the subject being thereby recalled to my notice, I found 
shortly afterwards the expression for the function 

a W + 4ac» + 463rf - 36M - Qabod 

(which cannot be expressed as a commutant) in the form of an intermutant, and 1 was thence leii 
to see the identity, so to say, of the theory of hyperdeterminants, as given in the memoir on 
Linear Transformations, with the present theory of intermutants. It is understood between Mr Sylvester 
and myself, that the publication of the present memoir is not to affect Mr Sylvester's right to 
claim the origination, and to be considered as the first publisher of such part as may belong to him 
of the theory hero sketched out. 



105] 



27 



105. 



CORRECTION OF THE POSTSCRIPT TO THE PAPER ON 

PERMUTANTS. 



[From the Cambridge and Dublin MathenuUical Journal, vol. vii. (1852), pp. 97 — 98.] 

Mr Sylvester has represented to me that the paragraph relating to his com- 
munications conveys an erroneous idea of the nature, purport, and extent of such 
communications; I have, in fact, in the paragraph in question, singled out what imme- 
diately suggested to me the expression of the function Qabcd + 36*0* — 400* — 46*^ — a'd* 
as a partial commutant or intermutant, but I agree that a fuller reference ought to 
have been made to Mr Sylvester's results, and that the paragraph in question would 
more properly have stood as follows: 

'* Under these circumstances Mr Sylvester communicated to me a series of formal statements, 

not only oral but in writing, to the effect that he had discovered a permutation method of obtaining 

SLA many invariants — viz. commutantive invariants — by direct inspection from a function of any degree 

of any nimiber of letters as the index of the degree contains even factors ; one of these conunu- 

tantive invariants being in fact the function ace-^-^hcd-ae^-bd^-i^^ expressible, according to Mr 

fa^ ah 6*\ 
Sylvester's notation, l>y ( 2' ^ jsl)'* ^^d, according to the notation of my memoir in the Camb. 



PhU. Tram., supposing 00= a, 01 = 10=6, 02 = ll = 20=c, &c. by 



00 
11 
22 



» 



Mr Sylvester and I shall, I have no doubt, be able to agree to a joint statement 
i>f any further correction or explanation which may be required. 



4—2 



28 



[106 



106. 



ON THE SINGULAKITIES OF SUEFACES. 



[From the Cambridge and Dublin Mathematical Journal, vol. vii. (1852), pp. 166 — 171.] 

In the following paper, for symmetry of nomenclature and in order to avoid 
ambiguities, I shall, with reference to plane curves and in various phrases and 
compoimd words, use the term •*node" as synonymous with double point, and the 
term "spinode" as synonymous with cusp. I shall, besides, have occasion to consider 
the several singularities which I call the "flecnode," the "oscnode," the "fleflecnode," 
and the "tacnode:" the flecnode is a double point which is a point of inflexion on 
one of the branches through it; the oscnode is a double point which is a point of 
osculation on one of the branches through it; the fleflecnode is a double point which 
is a point of inflexion on each of the branches through it; and the tacnode is a 
double point where two branches touch. And it may be proper to remark here, that 
a tacnode may be considered as a point resulting from the coincidence and amalga- 
mation of two double points (and therefore equivalent to twelve points of inflexion); 
or, in a different point of view, [?] as a point uniting the characters of a spinode and 
a flecnode. I wish to call to mind here the definition of conjugate tangent lines of 
a surfiEu;e, viz. that a tangent to the curve of contact of the surface with any 
circumscribed developable and the corresponding generating line of the developable, 
are conjugate tangents of the surface. 

Suppose, now, that an absolutely arbitrary surface of any order be intersected 
by a plane: the curve of intersection has not in general any singularities other than 
such as occur in a perfectly arbitrary curve of the same order; but as a plane can 
be made to satisfy one, two, or three conditions, the curve may be made to acquire 
singularities which do not occur in such absolutely arbitrary curve. 

Let a single condition only be imposed on the plane. We may suppose that 
the curve of intersection has a node; the plane is then a tangent plane and the 
node is the point of contact — of course any point on the surface may be taken for 



106] ON THE SINGULARITIES OF SURFACES. 29 

the node. We may if we please use the term "nodes of a surface," "node-planes of 
a surface/' as synonymous with the points and tangent planes of a surface. And it 
will be convenient also to use the word node-tangents to denote the tangents to the 
curve of intersection at the node; it may be noticed here that the node-tangents 
are conjugate tangents of the surface. 

Next let two conditions be imposed upon the plane: there are three distinct 
cases to be considered. 

First, the curve of intersection may have a flecnode. The plane (which is of 
course still a tangent plane at the flecnode) may be termed a flecnode-plane ; the 
flecnodes are singular points on the surface lying on a curve which may be termed 
the " flecnode-curve ^" and the flecnode-planes give rise to a developable which may 
be termed the flecnode-develope. The " flecnode-tangents " are the tangents to the 
curve of intersection at the flecnode; the tangent to the inflected branch may be 
termed the "singular flecnode-tangent," and the tangent to the other branch the 
"ordinary flecnode-tangent." 



Secondly, the curve of intersection may have a spinode. The plane (which is of 
course still a tangent plane at the spinode) may be termed a spinode-plane ; the 
spinodes are singular points on the surface lying on a curve which may be termed 
the "spinode-curve*." And the spinode-planes give rise to a developable which may 
be termed the " spinode-develope." Also the " spinode-tangent " is the tangent to the 
curve of intersection at the spinode. 

Thirdly, the curve of intersection may have two nodes, or what may be termed 
a "node-couple." The plane (which is a tangent plane at each of the nodes and 
therefore a double tangent plane) may be also termed a "node-couple-plane." The 
node-couples are pairs of singular points on the surface lying in a curve which may 
be termed the "node-couple-curve," and the node-couple-planes give rise to a deve- 
lopable which may be termed the " node-couple-develope." The tangents to the curve 
of intersection at the two nodes of a node-couple might, if the term were required, 
be termed the "node-couple-tangents." Also one of the nodes of a node-couple may 
be termed a " node-with-node," and the tangents to the curve of intersection at such 
point will be the " node-with-node-tangents." 

1 The fleonode-onrve, defined as the looas of the points through which can be drawn a line meeting the surface 
in four conseoaftive points, was, so far as I am aware, first noticed in Mr Salmon's paper **0n the Triple 
Tangent Planet of a Surface of the Third Order'* {Journal, t. iv. [1849], pp. 252—260), where Mr Salmon, 
unong other things, shows that the order of the surface being fi, the curve in question is the intersection of 
the surface with a surface of the order lln-24. 

* The notion of a spinode, considered as the point where the indicatriz is a parabola (on which account 
the spinode has been termed a parabolic point) may be found in Dupin's Diveloppements de OiomStrie : the 
most important step in the theory of these points is contained in Hesse's memoir "Ueber die Wendepuncte 
der Curren dritter Ordnnng" {CreUe, t. xxvin. [1848], pp. 97 — 107), where it is shown that the spinode-curve 
is the curre of intersection of the surface supposed as before of the order n, with a certain surface of the 
order 4(fi-2). See also Mr Salmon's memoir "On the Condition that a Plane should touch a surface along 
a Corre Line" {Jovrnal, t. in. [1848], pp. 44—46). 



30 ON THE SINGULARITIES OF SURFACES. [l06 

It is hardly necessary to remark that the flecnode-curve is not the edge of 
regression of the fleenode-develope, and the like remark applies m,fn. to the spinode- 
curve and the node-couple curve. 

Finally, let three conditions be imposed upon the plane: there are six distinct 
cases to be considered, in each of which we have no longer curves and developes, 
but only singular points and singular tangent planes determinate in number. 

First, the curve of intersection may have an oscnode. The plane (which continues 
a tangent plane at the oscnode) is an " oscnede-plane.'' The '' oscnode-tangents " are 
the tangents to the curve of intersection at the oscnode ; the tangent to the 
osculating branch is the " singular oscnode-tangent ; " and the tangent to the other 
branch the "ordinary oscnode-tangent." 

Secondly, the curve of intersection may have a fleflecnode. The plane (which 
continues a tangent plane at the fleflecnode) is a " fleflecnode-plane." The " fleflec- 
node-tangents " are the tangents to the curve of intersection at the fleflecnode. 

Thirdly, the curve of intersection may have a tacnode. The plane (which 
continues a tangent plane at the tacnode) is a " tacnode-plane.'' The ''tacnode- 
tangent" is the tangent to the curve of intersection at the tacnode. 

Fourthly, the curve of intersection may have a node and a flecnode, or what 
may be termed a node-and-flecnode. The plane (which is a tangent plane at the 
node and also at the flecnode, where it is obviously a flecnode-plane) is a "node-and- 
flecnode-plane." The " node-and-flecnode-tangents," if the term were required, would be 
the tangents to the curve of intersection at the node and at the flecnode of the 
uode-and-flecnode. The node of the node-and-flecnode may be distinguished as the 
node-with-flecnode, and the flecnode as the flecnode-with-node, and we have thus the 
terms " node-with-flecnode-tangents," " flecnode-with-node-tangents," " singular flecnode- 
with-node-tangent," and "ordinary flecnode- with-node-tangent." 

Fifthly, the curve of intersection may have a node and also a spinode, or what 
may be termed a " node-and-spinode." The plane (which is a tangent plane at the 
node, and is also a tangent plane at the spinode, where it is obviously a spinode-plane) 
is a " node-and-spinode-plane." The node-and-spinode-tangents, if the term were 
required, would be the tangents at the node and the tangent at the spinode of the 
node-and-spinode to the curve of intersection. The node of the node-and-spinode 
may be distinguished as the " nodo-with-spinode," and the spinode as the "spinode- 
Avith-node," and we have thus the terms " node-with-spinode-tangent," " spinode-with-node- 
tangent." 

Sixthly, the curve of intersection may have three nodes, or what may be termed 
a "node-triplet." The plane (which is a triple tangent plane touching the surfieu^ at 
each of the nodes) is a "node-triplet-plane." The "node-triplet-tangents," if the term 
were required, would be the tangents to the curve of intersection at the nodes of 
the node-triplet. Each node of the node-triplet may be distinguished as a "node- 



106 J ON THE SINGULARITIES OP SURFACES. 31 

with-node-couple," and the tangents to the curve of intersection at such nodes are 
•'node-with-node-couple-tangents." The terms " node-couple-with-node," '' node-couple-with- 
node-tangent," might be made use of if necessary. 

It should be remarked that the oscnodes lie on the flecnode-curve, as do also 
the fleflecnodes; these latter points are real double points of the flecnode-curve. The 
tacnodes are points of intersection and (what will appear in the sequel) points of 
contact of the flecnode-curve, the spinode-curve, and the node-couple-curve. The spinode- 
with-nodes are points of intersection of the spinode-curve and node-couple-curve, and 
the flecnode-with-nodes are points of intersection of the flecnode-curve and node-couple- 
curve; the node-with-node-couples are real double points (entering in triplets) of the 
node-oouple-curve. 

Consider for a moment an arbitrary curve on the surface; the locus of the node- 
tangents at the different points of this curve is in general a skew surface, which 
may however, in cases to be presently considered, degenerate in different ways. 

Reverting now to the flecnode-curve, it may be shown that the singular flecnode- 
tangent coincides with the tangent of the flecnode-curve. For consider on a surface 
two consecutive points such that the line joining them meets the surfistce in two 
points consecutive to the first-mentioned two points. The line meets the surface in 
four consecutive points, it is therefore a singular flecnode-tangent ; each of the first- 
mentioned two points must be on the flecnode-curve, or the singular flecnode-tangent 
touches the flecnode-curve. The two flecnode-tangents are by a preceding observation 
conjugate tangents. It follows that the skew surface, locus of the flecnode-tangents, 
brides up into two surfaces, each of which is a developable, viz. the locus of the 
singular flecnode-tangents is the developable having the flecnode-curve for its edge of 
regression, and the locus of the ordinary flecnode-tangents is the flecnode-develope. 
Of course at the tacnode, the tacnode-tangent touches the flecnode-curve. 

Passing next to the spinode-curve, the spinode-plane and the tangent-plane at a 
consecutive point along the spinode-tangent are identical^ or their line of intersection 
is indeterminate. The spinode-tangent is therefore the conjugate tangent to any other 
tangent line at the spinode, and therefore to the tangent to the spinode-curve. It 
follows that the surfEtce locus of the spinode-tangents degenerates into a developable 
sar£BU» twice repeated, viz. the spinode-develope. Consider the tacnode as two coin- 
cident nodes; each of these nodes, by virtue of its constituting, in conjunction with 
the other, a tacnode, is on the spinode-curve; or, in other words, the tacnode-tangent 
touches the spinode-curve, and the same reasoning proves that it touches the node- 
couple-curva It has already been seen that the tacnode-tangent touches the flecnode- 
curve ; consequently the tacnode is a point, not of simple intersection only, but of 
omtact, of the flecnode-curve, the spinode-curve, and the node-couple-curve. 

In virtue of the principle of the spinode-plane being identical with the tangent 
plane at a consecutive point along the spinode tangent, it appears that the tacnode- 

1 It most not be inferred that the tangent plane at rach oonBeeotiTe point is a spmode-plane ; thia ii 
obfTioody not the ease. 




32 ON THE SINGULARITIES OF SURFACES. [106 

plane is a stationary plane, as well of the flecnode-develope as of the spinode- 
develope, and it would at first sight appear that it must be also a stationary 
tangent plane of the node-couple-develope. But this is not so; the node-with-node- 
planes envelope, not the node-couple-develope, but the node-couple-develope twice 
repeated: the tacnode-plane is in a sense a stationary plane on such duplicate 
developable, but not in any manner on the single developable. The tacnode-plane is 
an ordinary tangent plane of the node-couple-develope. 

Consider now a spinode-with-node, which we have seen is a point of intersection 
of the spinode-curve and node-couple-curve. The tangent plane at a consecutive point 
along the spinode-with-node-tangent, is identical with the spinode-with-node-plane ; the 
curve of intersection of the tangent plane at such consecutive point has therefore a 
node at the node-with-spinode, or the tangent plane in question is a node-couple- 
plane, and the point of contact is a point on the node-couple-curve. Consequently 
the spinode-with-node-tangent touches the node-couple-curve, and thence also the 
spinode-with-node-plane is a stationary tangent plane of the node-couple-develope. 

It should be remarked that no circumscribed developable can have a stationary 
tangent plane except the tangent planes at the points where the curve of contact 
meets the spinode-curve, and any one of these planes is only a stationary plane 
when the curve of contact touches the spinode-tangent ; and that the node-couple- 
curve and the flecnode-curve do not intersect the spinode-curve except in the points 
which have been discussed 

Recapitulating, the node-couple-curve and the spinode-curve touch at the tacnodes, 
and intersect at the spinode-with-nodes : moreover, the tacnode-planes are stationary 
planes of the spinode-develope, and the spinode-with-node-planes are stationary planes 
of the node-couple-develope. Besides this, the two curves are touched at the tacnodes 
by the flecnode-curve, and the tacnode-planes are stationary planes of the flecnode- 
develope. 



107] 



33 



107. 



ON THE THEORY OF SKEW SURFACES. 



[From the Cambridge and Dublin Mathematical Journal, vol. vii. (1862), pp. 171 — 173.] 

A SURFACE of the n^ order is a surface which is met by an indeterminate line 
in n points. It follovrs immediately that a surface of the n^ order is met by an 
indeterminate plane in a curve of the n^ order. 

Consider a skew sur&ce or the surface generated by a singly infinite series of 
lines, and let the surfeice be of the n^ order. Any plane through a generating line 
meets the sur£Gtce in the line itself and in a curve of the (n — 1)^ order. The 
generating line meets this curve in (n — 1) points. Of these points one, viz. that 
adjacent to the intersection of the plane with the consecutive generating line, is a 
unique point ; the other (n — 2) points form a systenL Each of the (n — 1) points 
are svb modo points of contact of the plane with the surface, but the proper point 
of contact is the unique point adjacent to the intersection of the plane with the 
consecutive generating line. Thus every plane through a generating line is an ordinary 
tangent plane, the point of contact being a point on the generating line. It is not 
necessary for the present purpose, but I may stop for a moment to refer to the 
known theorems that the anharmonic ratio of any four tangent planes through the 
same generating line is equal to the anharmonic ratio of their points of contact, and 
that the locus of the normals to the sur&ce along a generating line is a hyperbolic 
paraboloid. Returning to the (n — 2) points in which, together with the point of 
contact, a generating line meets the curve of intersection of the sur&ce and a plane 
through the generating line, these are fixed points independent of the particular plane, 
and are the points in which the generating line is intersected by other generating 
lines. There is therefore on the surfSEU^ a double curve intersected in (n — 2) points 
by each generating line of the surfSsu^e — a property which, though insufficient to 
determine the order of this double curve, shows that the order cannot be less than 
(r - 2X (Thus for n = 4, the above reasoning shows that the double-curve must be 

an. 5 




34 ON THE THEORY OF SKEW SURFACES. [l07 

at least of the second order: assuming for a moment that it is in any case precisely 
of this order, it obviously cannot be a plane curve, and must therefore be two non- 
intersecting lines. This suggests at any rate the existence of a class of skew surfaces 
of the fourth order generated by a line which always passes through two fixed lines 
and by some other condition not yet ascertained; and it would appear that surfaces 
of the second order constitute a degenerate species belonging to the class in question.) 

In particular cases a generating line will be intersected by the consecutive 
generating line. Such a generating line touches the double curve. 

Consider now a point not on the surface ; the planes determined by this point 
and the generating lines of the surface are the tangent planes through the point; 
the intersections of consecutive tangent planes are the tangent lines through the 
point; and the cone generated by these tangent lines or enveloped by the tangent 
planes is the tangent cone corresponding to the point. This cone is of the n^ class. 
For considering a line through the point, this line meets the surface in n points, 
i.e. it meets n generating lines of the surface; and the planes through the line and 
these n generating lines, are of course tangent planes to the cone : that is, n tangent 
planes can be drawn to the cone through a given line passing through the vertex. 
The cone has not in general any lines of inflexion, or, what is the same thing, 
stationary tangent planes. For a stationary tangent plane would imply the inter- 
section of two consecutive generating lines of the surface. And since the number of 
generating lines intersected by a consecutive generating line, and therefore the number 
of planes through two consecutive generating lines, is finite, no such plane passes 
through an indeterminate point. The tangent cone will have in general a certain 
number of double tangent planes; let this number be x. We have therefore a cone 
of the class n, number of double tangent planes x, number of stationary tangent 
planes 0. Hence, if m be the order of the cone, a the number of its double lines, 
and fi the number of its cuspidal or stationary lines, 

7M = n (n — 1) — 2a?, 

/3 = 3w (w - 2) - 6x, 

a = in (n - 2) (n» - 9) - 2« (w» - n - 6) + 2« (a: - 1). 

This is the proper tangent cone, but the cone through the double curve is sub modo 
a tangent cone, and enters as a square factor into the equation of the general 
tangent cone of the order n (n — 1). Hence, if X be the order of the double curve, 
and therefore of the cone through this curve, 

m -»- 2X = n (ti - 1), and therefore X = x\ 

that is, the number of double tangent planes to the tangent cone is equal to the 
order of the double curve. It does not appear that there is anything to determine 
x\ and if this is so, skew surfaces of the v!^ order may be considered as forming 
dififerent families according to the order of the double curve upon them. 

To complete the theory, it should be added that a plane intersects the surface 
in a curve of the n*** order having x double points but no cusps. 



108] 



35 



108. 



ON CERTAIN MULTIPLE INTEGRALS CONNECTED WITH THE 

THEORY OF ATTRACTIONS. 



[From the Cambridge and Dublin Mathematicai Journal, vol. vii. (1852), pp. 174 — 178.] 

It is easy to deduce from Mr Boole's formula, given in my paper " On a Multiple 
Integral connected with the theory of Attractions," Journal, t. ii. [1847], pp. 219 — 223, 
[44], the equation 

df dv .^ f9-:it^ r «»-' {0x* -afds 






^/{(-S(-$)■! 



where n is the number of variables of the multiple integral, and the condition of the 
integration is 

(g -«.)•, (1? - ft y , =,. 

also where 

and e is the positive root of 

, (g-g.)' . (8-$,y ^ir 

0r jr + ^••■+7- 

€ + — e+ — 

Suppose /= jr... = ^i, and write (a — o,)' + ... = i', we obtain 

ii(?-«)*+-«^]»"-« f{in-q)r(q+ljj . (l+«)*» • 

5—2 



36 ON CERTAIN MULTIPLE INTEGRALS [108 

the limiting condition for the multiple integral being 

and the function a, and limit e, being given by 

€ denoting, as before, the positive root. Observing that the quantity under the integral 
sign on the second side vanishes for « = e, there is no difficulty in deducing, by a 
differentiation with respect to Ou the formula 



i[(f-«)'...+t^]*^"T(in-g)r(g)j. 



where (2S is the element of the surface (^--^1)' + ... = ^i', and the integration is 
extended over the entire surfieu^e, 

A slight change of form is convenient We have 

if we suppose 

The formulae then become 

r df... TT** r (g|V -h x^ - i;')g (fa 

i[(f'-«)"- + «^F"^ r(in-9)r(9 + l)J. «(! + «)**+« • 

in which e is the positive root of the equation 

I propose to transform these formula by means of the theory of images ; it will be con- 
venient to investigate some preliminary formulae. Suppose X* = o" + /S" . . . , V = «i* + A*. . . ; 
also consider the new constants a, 6,..., Oi, 61,..., m, /i, determined by the equations 

where S is arbitrary. Then, putting 



108] CONNECTED WITH THE THEORY OF ATTRACTIONS. 37 

it is easy to see that 



= V 






Proceeding to express the single integrals in terms of the new constants, we have in 
the first place A" = 8*4", where 






or if we write 

ooi + hhi ... = ZZi oos (k>, 

we have 



Hence also x~^3» where 

,• - A* jfc. "' 

whence 

._ 1 1 2%co8a> 

where p* = P + ii* — 2ZZi cos o), that is 

consequently ^jV + x* "" *^ = ^*n, where TI is given by 



n^ f' :^ iP'^^'-A') 



8 — 



and it is clear that e will be the positive root of 

It may be noticed that, in the particular case of ti = 0, the roots of this equation 
are 0, and — — J}^ '^^. Consequently if f^—fi^ and li^-fi are of opposite signs, 

we have 6 = 0; but \t f-f^^ and ii'-/i» are of the same sign, €=^^ •'-7^--'— ^ 



38 ON CERTAIN MULTIPLE INTEGRAX8 [108 

In order to transform the double integrals, considering the new variables x, y, ..., 
I write 45* + y'... =r" and 

whence also, if f' + i7*+ ... =p* (which gives rp = S^), we have 



a: = — — 

. I • • * 9 



also it is immediately seen tli*t 



(f-a)»+ ... «•= (ft ^^^.) ^ K^ - «)• + ••• +«*}. 

(f - a,)» ... -^,. = ^-^-^^{(a,-a,)»+ ... -/.'} : 

and from the latter equation it follows that the limiting condition for the first integral 
is (^— a,)'+... >/i' (there is no difficulty in seeing that the sign < in the former 
limiting condition gives rise here to the sign >), and that the second integral has to be 
extended over the surface (^ — 01)*+ ... =/i*. Also if dS represent the element of this 
surface, we may obtain 

did'n.,.^^dxdy ,.,, dX = ^^dS; 

and, combining the above formulae, we obtain 

r dxdy ... 

J (a^ + y« ... )*'»+« {(x - a)«-f (y- 6)' ... + i/'j*~-« 

"r(i^-9)r(j+i)(?+w«)*'»-^j. «(i+«)^+^' 

the limiting condition of the multiple integral being 

(^-a,)' + (y-6i)«...5/,«; 

and 

[ dS 

~ r ( jn - 9) r? (i« + 1*«)*'^ (/i> -/i>) J . (1 + «)***^^* ' 

where d^f is the element of the surface (a?— ai)* + (y — Ji)" ... =/i', and the integration 
extends over the entire surface. In these formute, I, li, p, 11 denote as follows: 

p = a«+6»+..., /i' = ai» + 6i«+..., p» = (a-ai)> + (6-6x)»+ ... , 
and € is the positive root of the equation 11 = 0. 



108] CONNECTED WITH THE THEORY OF ATTRACTIONS. 39 

The only obviously integrable case is that for which in the second fonnula q = l\ 
this gives 



/ 



dS 2w^f, 



(a^ + y» ...)*~ {(a?- a)> + (y- 6)« + tA>}*'*-» r(iw)(P + t4»)^»(Zi»-/i')(l +€)*~-' ' 



In the case of t* = 0, we have, as before, when p*— /i* and l^—fi are of opposite 
signs, € = 0, and therefore 1 -f € = 1 ; but when p* — /i* and i,' — /i' are of the same 
sign, the value before found for 6 gives 

1 + « = ^. i^/' + (p* -/>') (^' -/.*)}• 

Consider the image of the origin with respect to the sphere (a? — 01)*+ (y — 6i)"..^=/i', 
the coordinates of this image are 



> ••• » 



and consequently, if /a be the distance of this image from the point (a, 6 ...), we have 

/*'= {a- ^,(i.' -/.*)}»+.. . 



= ^ (^'/.' + (I'' -/x')W -/.')}; 



whence, by a simple reduction, 






or the values of the integral are 



/)•-/,» and k'-/i* opposite signs, ^ = f^^ F^^fc/O' 



n— 1 



p* -/.' and /,' -/,• the same sign, / = ^?^^ ___^^__ , 

where fi is the distance from the point (a, 6...) of the image of the origin with respect 
to the sphere (x — a^y + . . . — /i' = 0. 

Stone Buildinga, AvguM 6, 1850. 



40 



[109 



109. 



ON THE RATIONALISATION OF CERTAIN ALGEBRAICAL 

EQUATIONS. 



[From the Cambridge and Dublin Mathematical Journal, vol. viii. (1853), pp. 97 — 101] 

Suppose 

^ + y = 0, ic» = a, y* = 6 ; 

then if we multiply the first equation by 1, j^, and reduce by the two others, we havt^ 



from which, eliminating x, y. 



x+ y 


= 0. 


bx + ay = 0, 


I, 1 


= C 


b, a 





which is the equation between a and 6; or, considering x, y sa quadratic radicals, 

the rational equation between x, y. So if the original equation be multiplied by a-, //, 

we have 

a + xy-0, 

b + xy^O; 
or, eliminating 1, xy, 



a, 1 
6, 1 



= 0, 



which may be in like manner considered as the rational equation between x, y. 

The preceding results are of course self-evident, but by applying the same process 
to the equations 

ic + y + z = 0, a^ = ay y*^b, ^ = c, 



109] ON THE RATIONALISATION OF CERTAIN ALGEBRAICAL EQUATIONS. 41 



we have results of some elegance. Multiply the equation first by 1, yz, zx, xy, reduce 
and eliminate the quantities x, y, z, xyz, we have the rational equation 

111 =0; 
1 . c 6 
I c . a 
1 6 a . 

and again, multiply the equation by x, y, z, xyz, reduce and eliminate the quantities 
1, yz, zx, xy, the result is 

I a 6 c = 0, 
a . 1 1 
6 1.1 
ell. 

which is of course equivalent to the preceding one (the two determinants are in fact 
identical in value), but the form is essentially different. The former of the two forms 
is that given in my paper "On a theorem in the Geometry of Position" (Journal, 
voL II. [1841] p. 270 [1]): it was only very recently that I perceived that a similar 
process led to the latter of the two forma 

Similarly, if we have the equations 

X'\-y + z + w = 0, 5^ = a, ^" = 6, z^=^c, iv'^dy 

then multiplying by 1, yz, zXy xy, xw, yw, zw, an/zw, reducing and eliminating the 
quantities in the outside row. 



X. Vf '. 



tr, yzWf zwx, y>xy^ xyz 



we have the result 



1 1 1 


1 


• • • 


• 

1 
1 
1 


c 6 

c . a 
h a 


• 

• 
• 


1 . . 

. 1 . 
. . 1 


d . . 
. d . 
. . d 


a 
b 
c 


. 1 1 
1 . 1 
1 1 . 


• 
• 
• 


• • • 


• 


a b c 


d 



= 0; 



so if we multiply the equations by x, y, z, w, yzw, zwx, wxy, and xyz, reduce and 
eliminate the quantities in the outside row, 

C. II. 6 



42 



ON THE RATIONALISATION OF CERTAIN ALGEBRAICAL EQUATIONS. 



[109 



1, yz, zXt xy^ xto, yw, zwt xytw 



we have the result 



a 
h 
c 


. 1 1 
1 . 1 
1 1 . 


1 . . 
. 1 . 
. . 1 


• 
• 
• 


d 


• • • 


1 1 1 

c b 

c . a 
b a 


•> 

1 

1 
1 

1 


9 
• 
• 


d 

. d . 

. . d 


• 


a b c 


• • • 



= 0, 



which however is not essentially distinct from the form before obtained, but may be 
derived from it by an interchange of lines and columns. 

And in general for any even number of quadratic radicals the two forms are not 
essentially distinct, but may be derived from each other by interchanging lines and 
columns, while for an odd number of quadratic radicals the two forms cannot be so 
derived from each other, but are essentially distinct. 

I was indebted to Mr Sylvester for the remark that the above process applies to 
radicals of a higher order than the second. To take the simplest case, suppose 

^ + y = 0, a^-=a, y^=^b; 

and multiply first by 1, a^y, xy^\ this gives 

a? -f y . =0 



bx 



+ aj^y • = ; 



or, eliminating. 



1 1 



a 



1 

1 



-0; 



next multiply by x, y, a^\ this gives 

x" 



or, eliminating. 



. -\-xy = 
y« + iry = 
6a^ + ay* . =0; 

1 . 1 =0; 
. 1 1 
b a 



and lastly, multiply by aj*, y\ xy\ this gives 

b . + ary> = 
. ic*y + ay = ; 



109] ON THE RATIONAUSATION OF CERTAIN ALQEBRAICAL EQUATIONS. 43 



or, eUminating, 



a 1 . =0; 
6 . 1 
. 1 1 



where it is to be remarked that the second and third forms are not essentially distinct, 
since the one may be derived from the other by the interchange of lines and columns. 

Applying the preceding process to the sjrstem 

multiply first by 1, asyz^ c^}^s^^ ^z, y'j?, ah/^ ah/, y*^, s^x, reduce and eliminate the 

quantities in the outside row, 

«. Vt «» y'«'. ^V't y***» '^^y* ***'» *V 
the result is ~^_ I I =0; 



1 1 1 


• • • 


• • • 


• • • 


111 


• • • 


• • • 


• • • 


a b c 


. a 
b . . 
. c . 


1 . . 
. 1 . 
. . 1 


. 1 . 

. . 1 
1 . . 


. a . 
. . 6 
c . 


1 . . 
. 1 . 
. . 1 


. . 1 
1 . . 
. 1 . 



next multiply by a?, y, z, yV, s^a^^ a^y\ ahfz^ y*^a?, s^xy, reduce and eliminate the 
quantities in the outside row, 

the result is ""; I \ =0; 



1 . . 
. 1 . 
. . 1 


. 1 1 
1 . 1 
1 1 . 


• • • 

• • • 

• • • 


c 6 
c . a 
6 a 


• • • 

• • • 

• • • 


1 . . 
. 1 . 
. . 1 


• • • 

• • • 

• • • 


a . 
. 6 . 
. c 


. 1 1 
1 . 1 
1 1 . 



6—2 



44 ON THE RATIONALISATION OF CERTAIN ALGEBRAICAL EQUATIONS. [109 

lastly, multiply by a^, y*, z\ yz, zx, xy, xt/^s^^ ys^^> xyV^ reduce and eliminate the 
quantities in the outside row, 

1 xyz, zhjU\ yz\ «x», ary*, y*«, f'x, a?y 

the result is I" I = ; 



a 
b 
c 


• 
• 
• 




. 1 . 
. . 1 
1 . . 


. . 1 
1 . . 
. 1 


• 


1 
1 
1 




1 . . 
. 1 . 
. . 1 


1 . . 
. 1 . 
. . 1 




• 
• 

• 


1 
1 
1 


c 

a . 
. b . 


. 6 . 
c 
a 



where, as in the case of two cubic radicals, two forms, viz. the first and third forms 
of the rational equation, are not essentially distinct, but may be derived from each 
other by interchanging lines and columns. 

And in general, whatever be the number of cubic radicals, two of the three forms 
are not essentially distinct, but may be derived from each other by interchanging lines 
and columns. 



110] 



45 



110. 



NOTE ON THE TRANSFORMATION OF A TRIGONOMETRICAL 

EXPRESSION. 



[From the Cambridge and Dublin Mathematical Journal, vol. ix (1854), pp. 61 — 62.] 



The differential equation 

dx dy dz 



v = 0, 



(o + «) \/(c + a-") (a-^y)^{c + y) (a + z)>^(c + z) 

integrated so as to be satisfied when the variables are simultaneously infinite, gives 
by direct integration 

and, by Abel's theorem, 

1, X, (a H- a?) V(c + a?) =0. 

1, y, (a + y)V(c + y) 

, 1, z, (a + <2r) V(c + ^) ! 

To show d posteriori the equivalence of these two equations, I represent the deter- 
minant by the symbol Q, and expressing it in the form 

n = |lf a + x, (a + a:) V(c + a?) 



write for the moment f =. /f j &c. ; this gives 



46 NOTE OK THE TRANSFORMATION OF A TRIGONOMETRICAL EXPRESSION. [llO 



n = 



1. (a-c)(l + i). (o-c)»(^ + i) 



(o-c)'| f, f» + f p + 1 
(o-c)* P. f f + 1 



fVC» 






1. f. f 



-f^? 



1. e f 



} 



^ (a-c )»(g+iy + g-fi?g) 



fvr 



1. f f 



or, replacing (, i;, (f by their values, we have identically 

1, X, (o + a;) \/(c + a?) 
1, y, (p.+y)'J{c+y) 
1, z, (o + «)V(c+z) 

(c+a;)*(c+y)'(c+g)* f Ajc, / «— c la—c ja—c ja-—^ /"~^l 
(a - c)* IV €+«"*■ V c+y V c+z V c+«V c+yV c+rj 






— c a— c 



c+a?' C+J? 



— c a— c 



/a— c 



c+y' c+y 



— c a— c 



C+-8:' C+r 



and the equation 



/ g — c / g — c / g — c / g — c / g — c /g — c ^ ^ 
V c + a? V c + y V c + jgr Vc + a;V cTy V cT^ 



is of course equivalent to the trigonometrical equation 



tan- a/jHZ + tan-' a/^ + tan- a/^'' = 0. 
V c + « V c + y V c + £^ 



which shows the equivalence of the two equations in question. 



Ill] 



47 



111. 



ON A THEOREM OF K LEJEUNE-DIRICHLET'S. 



[From the Cambridge and Dublin Mathematical Journal, vol. ix. (1854), pp. 163 — 165.] 
The following formula, 

is given in Lejeune-Dirichlet's well-known memoir "Recherches sur diverses applica- 
tions &c." (Crelle, t. xxi. [1840] p. 8). The notation is as follows: — On the left-hand 
side (a, 6, c), (a', b\ c'), ... are a system of properly primitive forms to the negative 
determinant D (Le. a system of positive forms); x, y are positive or negative integers 
including zero, such that in the sum Sj*^"*'***^'^*''", aa^ + 2hxy 4- cy* is prime to 2jD, 
and similarly in the other sums ; q is indeterminate and the summations extend to 
the values first mentioned, of x and y. On the right-hand side we have to consider 
the form of jD, viz. we have D — PS' or else jD = 2PiSf, where S* is the greatest 
square factor in D and where P is odd: this obviously defines P, and the values 
of S, €, which are always ± 1 (or, as I prefer to express it, are always ±) are given 
as follows, viz. 

/) = PS*, P=l(mod4), S, €= + +, 

/> = PiS*, P = 3(mod4), S, € = -+, 

D = 2PiS*, P=l(mod4), S, € = + -, 

i> = 2P/S*, P = 3(mod4), S, € = , 

», n' are any positive numbers prime to 2Z), f pj is Legendre's symbol as generalized 
by Jacobi, viz. in general if /> be a positive or negative prime not a factor of n, 



48 ON A THEOREM OF M. LEJEUNE DIRICHLET's. [Hl 

then ( - j = + or — according as n is or is not a quadratic residue of p (or, what is 
the same thing, p being positive, f— j = w*<**"« (modp)), and for P=pp'p*' .,., 

and the summation extends to all the values of n, nf of the form above mentioned. 
In the particular case jD = — 1, it is necessary that the second side should be doubled. 
The method of reducing the equation is indicated in the memoir. The following are 
a few particular cases. 

i) = - 1, X^"^^ = 42 (-)*<»»-« q^^\ 

2) = — 2, V j«»+«y« = 22 ( — )* <**-" +* <***-*> g"*» , 

or (l + 25« + 25r«4-25r"...)(g + g* + g"4-g^ + ...) 

- 9 . _?•_ 2! £-_ + &c 

an example given in the memoir. 

i) = -3, 2?*"+'«^=22g)3**«', 

+ 2(5'* + ^''^ J^ 4- J»*^4- ...)(3* + g"' + ?~ 4- ?>~ ...) 

^g + g* g' + g* , g^ + g" g"-*-g " , 
l-V* 1-g" l-g« l-^"^*" 

I am not aware that the above theorem is quoted or referred to in any sub- 
sequent memoir on Elliptic Functions, or on the class of series to which it relates; 
and the theorem is so distinct in its origin and form from all other theorems relating 
to the same class of series, and, independently of the researches in which it originates, 
so remarkable as a result, that I have thought it desirable to give a detached state- 
ment of it in this paper. 



112] 



49 



112. 



DEMONSTRATION OF A THEOREM RELATING TO THE 

PRODUCTS OF SUMS OF SQUARES. 

[From the Philosophical Magazine, vol. iv. (1852), pp. 515 — 519.] 

Mb Ktrkman, in his paper " On Fluquatemions and Homoid Products of Sums of 
n Squares" {Phil, Mag. vol. xxxiii. [1848] pp. 447 — 459 and 494 — 509), quotes from a 
note of mine the following passage: — "The complete test of the possibility of the pro- 
duct of 2* squares by 2** squares reducing itself to a sum of 2^ squares is the following : 
forming the complete sjrstems of triplets for (2^ — 1) things, if eab, ecd, fac, fdh be any 
four of them, we must have, paying attention to the signs alone, 

(±eab)(±ecd) = (±f(w)(± fdh) ; 

Le. if the first two are of the same sign, the last two must be so also, and vice versd; 
I believe that, for a system of seven, two conditions of this kind being satisfied would 
imply the satisfsu^tion of all the others : it remains to be shown that the complete system 
of conditions cannot be satisfied for fifteen thinga" I propose to explain the meaning 
of the theorem, and to establish the truth of it, without in any way assuming the exist- 
ence of imaginary units. 

The identity to be established is 

(ti;* + o« -h 6» + ...) (w; + a; + b; ...) = w,; + a,; + b,; + ... 

where the 2* quantities w, a, 6, c, ... and the 2** quantities w,, a„ 6„ c,, ... are given quan- 
tities in terms of which the 2^ quantities w,^, a,,, 5,,, c,„ ... have to be determined. 

Without attaching any meaning whatever to the symbols a^, 6^, c^ ... I write down 

the expressions 

w + aa^ + bb^ + cc^..., w, + afl^ + bf>^ + c,c^ . . . , 

an. 7 



50 DEMONSTRATION OF A THEOREM RELATING [112 

and I multiply as if a^, 6^, c^... really existed, taking care to multiply without making 
any transposition in the order inter se of two symbols a^, 6^ combined in the way of mul- 
tiplication. This gives a quasi-product 

tvw, + (aw, + a,w) a^ + (btv, + b,w) 6^ + • • • 
+ aafl^* + bb,b^* + . . . 

-{-ab,aX + (^fibo<^o + "" 
Suppose, now, that a quasi-equation, such as 

means that in the expression of the quasi-product 

be, c a % d b , b , a c , b d 

are to be replaced by 

fl, 0, c, — ci,— 6, — c I 

*^o' o' o* o* o' o * 

and that a quasi-equation, such as ajb^c^ = — , means that in the expression of the quasi- 
product 

6c. cd. a b , c b , ac. b (i 

are to be replaced by 

-^o» ""^o> "-c^» ^o> Ky Co- 
It is in the first place clear that the quasi-equation, ajbjc^ = +, may be written in 
any one of the six forms 

a6c=4-, 6c(i=4", ca6=4-, 
a c b =^ —, c6a=5:— , ((20= — ; 

^0^0 o » *^o*'o o ' o**© o » 

and so for the quasi-equation ajb^c^ =» — . This being premised, if we form a system of 
(}uasi-equations, such as 

<^oMo = ±» ao^o«o = ±» &c- 

where the system of triplets contains each duad once, and once only, and the arbitrary 
signs are chosen at pleasure; if, moreover, in the expression of the quasi-product we 
replace a,', 6^', ... each by — 1, it is clear that the quasi-product will assume the form 

^/y» ^y/» K* C// ••• b^iiig determinate functions of w, a, b, c, ...; w,, a,, b„ c, ..., homo 
geneous of the first order in the quantities of each set ; the value of w,, being obviously 
in every case 

w,^ = vm^ — tta, — 66, — cc, . . . , 

and a,,, b„, c,,,... containing in every case the terms aw,-{-a,w, btv, + b,w, cw,-¥ cjw,.,, but 
the form of the remaining terms depending as well on the triplets entering into the 



112] TO THB PRODUCTS OF SUMS OF SQUARES. 51 

system of quasi-equations as on the values given to the signs ± ; the qtiosi-egtuxtions 
serving, in fact^ to preset^ a rule for the formation of certain functions w,„ a^„ 6,,, c,,, ..., 
the properties of which functions may afterwards be investigated. 

Suppose, now, that the system of quasi-equations is such that 

e a b , e c d 



O O O' "O^O O 



being any two of its triplets, with a common symbol e^, there occur also in the system 

the triplets 

f fi c , f d b , Odd, a b c \ 

and suppose that the corresponding portion of the system is 

^o^o^o = €, e^c^d^ = c', 
/o«oCo = (r, fodoK = ^» 

where €, f, *, e', ^, t each of them denote one of the signs + or — ; then e,„f„, g,, will 
contain respectively the terms 

€ (ab, - a,b) + e' (cd, - c^d), 

i (ad^ — a,d) + if (be, — bfi) ; 

and e,,^+f*+g,* contains the terms 

(a> 4- 6» + c» + d«) (a; + b; + c/ + d;) - aJ'a; - H; - cFc^-dl'd; 

+ 2 [ee {ab, - ajb) (cd, - c,d) 
+ ^(ac,-afi){db,-'dfi) 
+ u' {ad, - a,d) (be, - bfi)] ; 

and by taking account of the terms ew, + e,w, fw,+f,w, gw,-\-g,w in e„, f,„ gr^/ respect- 
ively, we should have had besides in ^^ -^-f^ + g^ ^® terms 

-\-2(ee,+ff^ + gg,)ww,. 

Also k;^' contains the terms 

ii/Hv,*-\'a*a,^ + b'b; + (^; + dM,^ 
-i(ee,+ff,+gg,)ww,] 



whence it is easy to see that 



(ti;* + a« + 6« + c*+ ...)(w/ + a/ + V + c/ + ...) 

+ 2S [ee' (a&, - ajb) (cd, - c^d) 

+ ^(ac,-a,c)(db,-d,b) 

+ ii' (ad, — a,d) (be, — 6^c)]. 

7—2 



52 DEMONSTRATION OF A THEOREM RELATING &€. [112 

where the summation extends to all the quadruplets formed each by the combination 
of two duads such as ab and cd, or ac and db, or ad and be, L e. two duads, which, com- 
bined with the same common letter (in the instances just mentioned e, or /, or g), enter 
as triplets into the sjrstem of quasi-equations — so that if i/ = 2** — 1, the number of quad- 
ruplets is 

i{H''-l)i(''-3)}v.i, = Av(.;- !)(,;- 3). 
and the terms under the sign 2 will vanish identically if only 

but the relation e^ = a' is of the same form as the equation cc' = ff ; hence if all the 
relations 

are satisfied, the terms under the sign S vanish, and we have 

{w^; 4- a^; + 6,/ + c,;+ ...) = («^ + a' + 6" + c«+ ...) (w; + a; + 6/ + c/ + ...) 
which is thus shown to be true, upon the suppositions — 

1. That the sjrstem of quasi-equations is such that 

6 db ^ 6 c d 



being any two of its triplets with a common sjrmbol e^, there occur also in the sjrstem 
the triplets 

2. That for any two pairs of triplets, such as 

e a b , 6 c d and f a c , f d b , 

the product of the signs of the triplets of the first pair is equal to the product of the 
signs of the triplets of the second pair. 

In the case of fifteen things a, 6, o, ... the triplets may, as appears from Mr Kirk- 
man's paper, be chosen so as to satisfy the first condition; but the second condition 
involves, as Mr Kirkman has shown, a contradiction; and therefore the product of two 
sums, each of them of sixteen squares, is not a sum of sixteen squares. It is proper to 
remark, that this demonstration, although I think rendered clearer by the introduction of 
the idea of the system of triplets furnishing the rule for the formation of the expres- 
sions w,,, a,,, 6,„ c^,, &a, is not in principle different fi-om that contained in Prof. Young's 
paper "On an Extension of a Theorem of Euler, &c.", Irish Transactions, vol. xxi. [1848 
pp. 311—341]. 



113] 



53 



113. 



NOTE ON THE GEOMETRICAL REPRESENTATION OF THE 

INTEGRAL jdx^ >/(x + a) ~(aj + h){x + c). 



[From the Philosophical Magazine, voL v. (1863), pp. 281 — 284.] 

The equation of a conic passing through the points of intersection of the conies 

a^ + y* + ^ = 0, 

is of the form 

w (ic* + y* + 2:*) + cut" + 6y« + ag« = 0, 

where k; is an arbitrary parameter. Suppose that the conic touches a given line, we 
have for the determination oi w b, quadratic equation, the roots of which may be 
considered as parameters for determining the line in question. Let one of the values 
oi w he considered as equal to a constant quantity A;, the line is always a tangent 
to the conic 

*(«" + y* + ^) + flwj* + 6y' + C4^ = ; 

and taking w^p for the other value of t^;, /) is a parameter determining the parti- 
cular tangent, or, what is the same thing, determining the point of contact of this 
tangent 

The equation of the tangent is easily seen to be 

X ^h-c '/a + k VoTp + y'</c — a^b + k '^b+p-^-z 'Ja-h "Jc + k '^cTp^ ; 

suppose that the tangent meets the conic a^ + y^+z^ = (which is of course the 
conic corresponding to w = oo ) in the points P, P', and let d, oo be the parameters 
of the point P, and ^, oo the parameters of the point P', i.e. (repeating the defini- 



54 NOTE ON THE GEOMETRICAL REPRESENTATION OF [ll3 

tion of the terms) let the tangent at P of the conic a^H-y' + £^ = be also touched 
by the conic d(a^ + y' + ^') + aa?» + 6y» + 0^ = 0, and similarly for ff. The coordinates 
of the point P are given by the equations 

x:y:z=^ 'Jb-c Va + d: ^c-a 'JbTO : Va-6 Vc + tf ; 
and substituting these values in the equation of the line PP", we have 

(b - c) Va + A VoTp Va + d + (0-0) ^bTk y/b+p ^/W^ + (a - 6) \/c^k y/c+p Vc-f (? 

= 0. ..(♦), 

an equation connecting the quantities p, 0. To rationalize this equation, write 

V(a + i) (a +p)~(a + d)~= X + /ui, 
V(6 4-A)(6+i>)(6 + d) = X + ^, 

values which evidently satisfy the equation in question. Squaring these equations, we 
have equations from which X', \p^ fj} may be linearly determined ; and making the 
necessary reductions, we find 

X'* = a6c + kpd, 
- 2X/i= 6c + ca + a6-(p^4-Ap4- kd\ 

or, eliminating X, /[t, 

{6c + ca + a6-(p5 + ip4-A;d)}'-4(a4-6 + c + A?+p + d)(a6c + A:/>d) = 0, (♦), 

which is the rational form of the former equation marked (*). It is clear from the 
symmetry of the formula, that the same equation would have been obtained by the 
elimination of Z, M from the equations 



V (jfc + a) (A + 6) (Jfc + c) = Z + Mk, 
V(p + a)(/> + 6)(jp + c) ^L + Mp, 



and it follows from Abel's theorem (but the result may be verified by means of 
Euler's fundamental integral in the theory of elliptic functions), that if 

dx 



J • V(a! + o) (a + 6)(a! + c) ' 



then the algebraical equations (*) are equivalent to the transcendental equation 

±m±nj)±nd=0; 



113] THE i^n:m}RAh jdx-^J{x + a){x + b){x + c). 55 

the arbitrary constant which should have formed the second side of the equation 
having been determined by observing that the algebraical equation gives for p^d, 
& = 00 , a system of values, which, when the signs are properly chosen, satisfy the 
transcendental equation. In fact, arranging the rational algebraical equation according 
to the powers of i, it becomes 

*»(p - ey - 2k {p^(p + ^) + 2 (a + 6 4-c)pe + (fcc + ca 4- ah){p + ^) + 2abc] 
-\'jfe'''2{bc-{-ca'\-ah)pd'' 4a6c (jp + d) + fr^c* + c»a» + a^6' - 2a^bc - 26«ca - 2c^ab = ; (♦) 

which proves the property in question, and is besides a very convenient form of the 
algebraical integral. The ambiguous signs in the transcendental integral are not of 
course arbitrary (indeed it has just been assumed that for p = 0y Up and RO are to 
be taken with opposite signs), but the discussion of the proper values to be given 
to the ambiguous signs would be at all events tedious, and must be passed over for 
the present. 

It is proper to remark, that d=p gives not only, as above supposed, &=x, 
but another value of k, which, however, corresponds to the transcendental equation 

±Uk± 2np = ; 

the value in question is obviously 

, _ fi* - 2 (6c+ eg + ab)p^ - Sahcp + 6V + Ca' + a'6^ ~ 2a»6c - 2fe'co- 2(^06 
"" (p + a) (p + 6) (|) + c) 

Consider, in general, a cubic function aai^ + Sba^y + 3ca:y* 4- dy*, or, as I now write 
it in the theory of invariants, (a, b, c, d) {x, yY, the Hessian of this function is 

(ac-b», Had-bc), bdr.c>)(a?, y)>, 

and applying this formula to the function (p 4- a) (p 4- 6) (p 4- c), it is easy to write 
the equation last preceding in the form 

lA'- D (a\b\c^ ^ Se8aiB,n {(p + a)(p 4- b)(p 4- c)} 
^^~^ (a4-64-c) (p4-a)(p4-6)(p + c) 

which is a formula for the duplication of the transcendent Ilx, 
Reverting now to the general transcendental equation 

±nA:±IIp±nd = 0, 
we have in like manner 

±ni±np±n^=0; 

and assuming a proper correspondence of the signs, the elimination of Tip gives 



56 NOTE ON THE GEOMETRICAL REPRESENTATION &C. [ll3 

i.e. if the points P, P* upon the conic a;* + y' + 2r* = are such that their parameters 
6, ff satisfy this equation, the line PP will be constantly a tangent to the conic 

k {x^ + y' 4- z") + {aa? + 6y« + c^) = 0. 

Hence also, if the paraibeters ^^ k\ Id' of the conies 

k (aJ» + y' + -2») + cw^ + 6y* + c^ = 0, 
k' (a^ + y' + -^*) + cur» + 6y»+C4:» = 0, 
A"(«* + y» + ^«) + (m;» + 6y> + c^» = 0, 



satisfy the equation 



nA: + nA' + nr = o, 



there are an infinity of triangles inscribed in the conic a;* + y' + -j* = 0, and the sides 
of which touch the last-mentioned three conies respectively. 

Suppose 2nA; = II/c (an equation the algebraic form of which has already been 
discussed), then 

= CO gives ff ^ K\ or, observing that ^ = oo corresponds to a point of intersection 
of the conies a^ + y" + £^ = 0, cw^ 4- 6y' + C2^ = 0, k is the parameter of the point in 
which a tangent to the conic A:(a;' + y* + -»®) + a^ + 6y' + c^ = at any one of its 
intersections ¥dth the conic aJ'-hy* + 4:* = meets the last-mentioned conic. Moreover, 
the algebraical relation between 0, ff and k (where, as before remarked, /v is a given 
function of k") is given by a preceding formula, and is simpler than that between 
d, ff and k. 

The preceding investigations were, it is hardly necessary to remark, suggested by 
a well-known memoir of the late illustrious Jacobi, and contain, I think, the extension 
which he remarks it would be interesting to make of the principles in such memoir 
to a system of two conies. I propose reverting to the subject in a memoir to be 
entitled "Researches on the Porism of the in- and circumscribed triangle." [This was, I 
think, never written.] 



114] 



57 



114. 



ANALYTICAL RESEARCHES CONNECTED WITH STEINER'S 

EXTENSION OF MALFATTI'S PROBLEM. 



[From the Philosophical Transaciiona of the Royal Society of London, vol. CXLII. for the 
year 1852, pp. 253—278: Received April 12,— Read May 27, 1852] 

The problem, in a triangle to describe three circles each of them touching the two 
others and also two sides of the triangle, has been termed after the Italian geometer 
by whom it was proposed and solved, Malfatti's problem. The problem which I 
refer to as Steiner^s extension of Malfatti's problem is as follows: — "To determine 
three sections of a surface of the second order, each of them touching the two others, 
and also two of three given sections of the surface of the second order," a problem 
proposed in Steiner's memoir, "Einige geometrische Betrachtungen," CreUCy t. i. [1826 
pp. 161 — 184]. The geometrical construction of the problem in question is readily 
deduced from that given in the memoir just mentioned for a somewhat less general 
problem, viz. that in which the surface of the second order is replaced by a sphere ; 
it is for the sake of the analytical developments to which the problem gives rise, that 
I propose to resume here the discussion of the problem. The following is an analysis of 
the present memoir: — 

§ 1. Contains a lemma which appears to me to constitute the foundation of the 
aaalytical theory of the sections of a surfietce of the second order. 

§ 2. Contains a statement of the geometrical construction of Steiner's extension 
of Mal£Eitti's problem. 

§ 3. Is a verification, founded on a particular choice of coordinates, of the con- 
struction in question. 

§ 4. In this section, referring the surface of the second order to absolutely general 
coordinates, and after an incidental solution of the problem to determine a section 
touching three given sections, I obtain the equations for the solution of Steiner's 
^tension of Malfisitti's problem. 

an. 8 



58 ANALYTICAL RESEARCHES CONNECTED WITH [ll4 

§ 5. Contains a separate discussion of a sjrstem of equations, including a^ a 
particular case the equations obtained in the preceding section. 

^ 6 and 7. Contain the application of the formulae for the general system to the 
eijuations in § 4, and the development and completion of the solution. 

§ 8. Is an extension of some preceding formulae to quadratic functions of any 
number of variables. 



§ 1. Lenwia relating to the sections of a surface of the second order. 

If 

flWJ" + fry* + c^ + dtu>' + 2fyz -h 2gzx + 2hxy + 2lxw -h 2mt/w -h 2nzw = 

be the equation of a surface of the second order, and 

the reciprocal equation, the condition that the two sections 

\'x -h fiy -h v'z -h p^w = 0, 
may touch, is 

(ax* -h iSfi^ + ©!/» + Bp' -h 2JpMi' + 2(Sv\ + i'f^XfjL + 2%\p -h i^fip -h 2^vp)^ 

X (av» + i8^'» + €v' + B/» + 2jffiV + 2(SvX + 2f^\v + 2'a\y -h 2iW/tv + 2^v'py 

+ U(Xp' -h \'p)-hMW +M» + ia (vp'-^v'p)): 
and in particular if the equation of the surface be 

cux^ -\- bt/* -\- cz^ -\- 2fyz -h 2gzx -h 2hay-\-pvj^ = 0, 
the condition of contact is 

ax« +i8/i» +ffii/» +2JP/IAI/ -h2CBi/X+2|^X^-h-p»j 



(ax'« +i8M« +€^'' +2JF /iV + 2flri.V + 2?^XV + ^p 



in which last formula 

a=6c-/', 13*oa>-5r», aD=o6-A», 
§ = gh-af , (S = 1if-bg. ^=fg-ch, 
K = ahc - of* - bf - ch* + 2/gh. 



114] steimer's extension of malfatti's pboblem. 59 

§2. 

In order to state in the most simple form the geometrical construction for the 
solution of Steiner's extension of Malfatti's problem, let the given sections be called 
for conciseness the determinators ^ ; any two of these sections lie in two different cones, 
the vertices of which determine with the line of intersection of the planes of the 
determinators, two planes which may be termed bisectors ; the six bisectors pass three 
and three through four straight lines ; and it will be convenient to use the term 
bisectors to denote, not the entire system, but any three bisectors passing through the 
same line. Consider three sections, which may be termed tactors, each of them touching 
a determinator and two bisectors, and three other sections (which may be termed 
separators) each of them passing through the point of contact of a determinator and 
tactor and touching the other two tactors ; the separators will intersect in a line which 
passes through the point of intersection of the determinators. The three required 
flections, or as I shall term them the resultors, are determined by the conditions that 
each resultor touches two determinators and two separators, the possibility of. the 
ooDstruction being implied as a theorem. The d posteriori verification may be obtained 
as follows: — 

§ 3. 

Let J? = 0, y = 0, « = be the equations of the resultors, w; = the equation of the 
polar of the point of intersection of the resultors. Since the resultors touch two and 
two, the equation of the surfisice is easily seen to be of the form 

2yz + 2zj? -h 2xy + k/« = 0. («) 

The determinators are sections each of them touching two resultors, but otherwise 
arbitrary; their equations are 

-ax + ^^y + ^z-^w^O, 

The separators are sections each of them touching two resultors at their point of 
contact (or what is the same thing, passing through the line of intersection of two 
resultors), and all of them having a line in common. Their equations may be taken 
to be 

cy — 6-2: = 0, cw^ — ca? = 0, 6a? - ay = 0, 

M use the words ** detenninatorB,'' <ftc. to denote indifferently the sections or the planes of the sections; 
the context is always saffident to prevent ambignity. 
* The reeiprooal form is, it should be noted, 

8—2 



60 ANALYTICAL RESEABCHES CONNECTED WITH [114 

the values of a, 6, c remaining to be determined. Now before fixing the values of 
these quantities, we may find three sections each of them touching a detenninator at 
a point of intersection with the section which corresponds to it of the sections 
cy — 6-? = 0, aZ'-ca) = 0, for — ay = 0, and touching the other two of the last-mentioned 
sections ; and when a, 6, c have their proper values the sections so found are the 
tactors. For, let \x-\-fMy-\-vz + f>w = be the equation of a section touching the deter- 

minator — air-h^y-h^^ + t£; = 0, and the two sections &c — ay = 0, az — cx = 0: and 

suppose 

A« = X« + m' + »^ - 2/Av - 2i/X - 2X/I - 2p» ; 

the conditions of contact with the sections 6a? — ay = 0, ae — ca? = are found to be 

(6 4- a) A = (6 + a) \ - (6 + a) ft - (6 - a) I/, 
(c + a) A = (c + a) X — (c — a) fi - (c + a) I/, 

values, however, which suppose a correspondence in the signs of the radicals. Thence 
(b -\- a) fi = (c + a)v ; or since the ratios only of the quantities \ fi, v, p are material, 
fi = c '\' a, j^ = 6 + a, and therefore 

A« = X« - 2 (2a -h 6 4- c) X + (6 - cy - 2p«, = (X - 6 - cf, 

or p» = — 2 (aX + 6c). 

Hence the equation to a section touching 6a? — ay = 0, a^ — ca? = is 

Xj?4-(c + a)y + (6 + a)ir-hV -2(aX + 6c)\ w = ; 

and to express that this touches the detenninator in question, we have 

± a (X - 6 - c) = (a + -) X - a (2a + 6 -h c) + 2 V-2(aX + 6c) ; 

and selecting the upper sign, 

1 J 

- X-2aa = -2V-2(aX + 6c); 

whence 

X = - 2a {aa-^ " 26c), V - 2 (aX -h 6c) = (2aa - V ^6c) ; 

or the section touching the determinator and the sections 6a? — ay = 0, a-? — ca? = is 

-2a(aa- V-26c)«r-h(c + a)y + (6-ha)-? + (2aa->/-26c)w = 0; 
and at the point of contact with the determinator 

2y« + &?« + 2ay + «;• = 0. 



114] steiner's extension of malfatti's problem. 61 

Eliminating w between the first and second equations and between the second and 
third equations, 

V-26c (ax -h - y 4- — ^^ + cy + 6-? = 0, 

and from these equations (cy — bzy = 0, or the point of contact lies in the section 
cy — bz^O. It follows that the equations of the tactors are 

-2a(aa-V-26c)a? + (c + a)y + (6 + a)-8:+(2aa- V' - 26c) w = 0, 
(c + 6) a: - 2/8 (6/8 - V - 2ca) y + (a + 6) -? + (26/8 - V - 2ca) w = 0, 
(6 + c) a: + (a + c) y - 27 (07 - V- 2a6) -? -h (2c7 - V-2a6) w = 0, 
where a, b, c still remain to be determined. 

Now the separators pass through the point of intersection of the determinators ; 
the equations of these give for the point in question, 

X : y : z : «; = (2/87 -h 1) (- a + /8 + 7 + 2a^7) 

(27a +1)( a-/8 + 7 + 2a/87) 

(2a/3 + l)( a4-/3-7 + 2a/37) 

4a»/8V-l+a» + /9» + 7«; 

and the values of a, 6, c are therefore 

a :b : c = (2/87+ 1) (-a + i8 + 7 + 2a/37) 
:(27a+l)( a - /8 + 7 + 20/87) 
:(2a/8 + l)( a + /8-7 + 2a/87), 

which are to be substituted for a, 6, c in the equations of the separators and tactors 
respectively. 

Now proceeding to find the bisectors, let Tix + fiy + vz + pw^O be the equation 
of a section touching the determinators, 

^«-/3y + ^* + w-0. ^x + ^y-yi + w = 0; 

and suppose, as before, A**X*-hfi* + i^ — 2/ij^ — 2i/X — 2X/i — 2p*; the conditions of con- 
tact are 



±/8A = /3X-(/8 + i)/i + /8i/-2p, 



T 7A = 7X + 7/1 — f 7 + - j V — 2p, 



62 ANALYTICAL RESEARCHES CONNECTED WITH [114 

where it is necessary, for the present purpose, to give opposite signs to the radicals. 
For if the radicals had the same sign, it would follow that 

^|^i8\-(/3 + ^)M-h/3i/--2pJ-^[^7X-h7/i-(7H-^),;-2pj=0; 

hence the section \x + fiy -{- vz -\- pw =^ would pass through the point 

nil 22 

or the section would be a tangent section of the two determinators of the same 
class with the resultor a? = 0, which ought not to be the case. The proper formula is 

^|^/3X-(^ + ~)AiH-/3i/-2p]+^[^7\ + 7/i-(7 + ^)i'-2p]=0; 

and this equation being satisfied, the section 

passes through a point 

«o 112 2 

a: : y . z : w -- z : -gi^-y'-^"*- 

The bisector passes through this point and the line of intersection of the determi- 
nators ; its equation is 



l^x-fiy + lz + wy^{^x + ^y-yz + w)=0; 






or reducing and completing the system, the equations of the bisectors are 

In order to verify the geometrical construction, it only remains to show tliat 
each bisector touches two tactors. Consider the bisector and tactor 

-(^ + 2^)«+(l + 2^)y + (2^-2|')' + (-«-|)«' = ^' ■ 
- 2a (aa - V-26c) ar + (c + a)y-h(6-ha)-^ + (2aa- V-2ic) w = 0; 

and represent these for a moment by 

Xa? -h fiy + 1/-? + /Mi; = 0, X'a? + //y + p'z + p'w = ; 



114] steiner's extension of malfatti's problem. 63 

if A be the same as before, and A' the like function of \\ ja\ v\ p, also if 

* = XX' + /i/A' -h w' - (/lAi/' + fi'v) - (vV + v\) - (X/ + X» - 2pp', 



then 



-^■-(^^s)"' 



A'* = (200? - 2a V - 26c + 6 + c)», 

and the condition of contact AA' = <I> {taking the proper rign for the radicals) be- 
comes 



or reducing, 



"*-^^ + ''2^1 = ^' 



an equation which is evidently not altered by the interchange of a, a and 6, fi. The 
conditions, in order that each bisector may touch two tactors, reduce themselves to 
the three equations, 

-*'+*2-^i'^'^=<*' 

which are satisfied by the values dbove found £nr tk^ quantities a, h, c. The possi- 
hility and truth of the geometrical construction are thus demonstrated. 

Let it be in the first instance proposed to find the equation of a section touch in^^ 
all or any of the sections a? = 0, 3r = 0, £ = of the surfi^^ of the second order, 

^wc" + 6y" + ci^ + Sfyz + 2gzx -h thxy + pw?'^ = 0. 
Any section whatever of this surface may be written in the form 

(aX + Afi + gv) x + (h\ + i/i -h /i/) y + (g\ -{-ffi + ci/) ^ + V — ;> V w = 0, 

where 

V« =:aX«+ V H-cr' + 2//iav + igvX -f 2A\/i - K, 



64 ANALYTICAL RESEARCHES CONNECTED WITH [ll4 

and \ fi, V are indeterminate. And considering any other section represented by a 
like equation, 

{pX + hfi'-\-gi/) X -h (AX' -h bfi' +>') y + (flfV +// + cv') z + V^ V'w = 0, 

where 

V'« = aV» + 6/» + ci;'» + 2//j/' + 2fln^ V + 2A\y - K, 

it may be shown by means of the lemma previously given, that the condition of 
contact is 

aXX' + 6/a/a' + cvv' -\-f(p,v' + iLv) + g (i/X' + v\) + A (X/ia' + X» ± iT = VV. 
Suppose that X', y!^ v' satisfy the equations 

AV + 6/+yj/' = 0, 
gfX' +// + CI/' = 0, 

so that the last-mentioned section becomes a; = 0; and observing that the first of 
the above equations may be transformed into 

/» 1^ CBf 

it is easy to obtain X' = v W, /*' = "Tg , v' = j^ . The condition of contact thus becomes 

K 



and taking the under sign, X = a/^I, so that if in the above written equation we 

establish all or any of the equations X = V^, fi = V3B, v = V®, we have the equation 
of a section touching all or the corresponding sections of the sections 

«; = 0, y = 0, z = 0. 

In particular we have for a solution of the problem of tactions, the following 
equation of the section touching a? = 0, y = 0, jf = 0, viz. 



+ — . 



Anticipating the use of a notation the value of which will subsequently appear, 
or putting 

f=^l^/vgB®-JF, g=-ys7vH3ffi; h = ^Vvi3B-|^, j=V24'gti3ar, 



114] steiner's extension of malfatti's problem. 65 

values which give 

ir« = -f*-g*-h* + 2g»h« + 2h«f« + 2f»g«-^^', 

the equation of the section in question is 

7^<-f^ + g'-hh')a:-h;^(P-g'-hh')y-h^(P-hg'~h')^-h 'g^''/'''' i(; = 0. 

I proceed to investigate a transformation of the equation for the section with an 
indeterminate parameter X, which touches the two sections y = 0, z=^0. We have 

aV» = (oX + V + 9^y + (^/^^ + i8i/" - 2 jp/ii/) - J8ffi + Jp ; 

or putting for ^ and v their values VS, V(!D in the second term, 

aV'* = (aX + A/A + 5ri/)» + (VS® - Jp)» ; 

and introducing instead of X an indeterminate quantity X, such that 

aX + A/i + (7J^ = (VS®-jp)Z, 
we have 

also introducing throughout X instead of X, and completing the substitution of V33, v CD, 
for /A, V, the equation of the section touching y = 0, ^r = 0, becomes 

{aX'^hy'\'gz)X'k-y V® + -^ V:JB + w V -op Vl -h Z» = 0. 

It may be remarked here in passing, that this is a very convenient form for the 
demonstration of the theorem; "If two sections of a surface of the second order touch 
each other, and are also tangent sections (of the same class) to two fixed sections, 
then considering the planes through the axis of the fixed sections and the poles of 
the tangent sections, and also the tangent planes through this axis, the anharmonic 
ratio of the four planes is independent of the position of the moveable tangent sections;" 
where by the axis of the fixed sections is to be understood the line joining their poles. 

The sections touching -? = 0, a?=0, and iF = 0, y = 0, are of course 

x^-k-Qix + hy -\'fz) Y-\-z Va + w V-fcpVl-f r« = 0, 

a? VS +y Va + (flra; -h/y + c-?) Z + w "J ^cp Vl +i^''= 0, 
where 

Ax' +6/i' +/i;' =(V^a-<!B)7, X' = Vir / =/. »'' = V®, 

The conditions of contact of the sections represented by the above written equations 
would be perhaps most simply obtained directly from the lemma, but it is proper to 
deduce it firom the formula for contact used in the present memoir. If for shortness 

<I>(±) = aX'X" + 6/iV' -h cv V +/(/!/" -h Ai'V) + g {vX' + i/'V) + A (XV + X'V) ± K. 
C. II. 9 



66 ANALYTICAL BBSEABCHES CONNECTED WITH [114 

where the symbol 4>(±) is used in order to mark the essentially different character 
of the results corresponding to the different values of the ambiguous sign, then 

bap (_) =/(Ax' + V +>') (3^" +//*" + <»"). 
+ 0SLv' - «SX' ) (g\" +/fi," + cp"). 
+ (a/' - I^X") ihX' + 6/ +>' ). 
+ v'fi" i-fSf)+ v'\"f^ + xy/ffi + XV (K -f$) 

= /(AX' + bfi,' +Jv') ig\" ^fy!' + cv") 
+ Va (Va® - <S) O^X" +//*" + (»") 
+ Va ( V^S - IQ) (AX' + hy! + >' ) 

+/C- a ^s®'+ 1^ v®if + ffi VI0B - a jF - («5?^ - a jF)) 

= /(AX' + 6/*' +/;') (jrX" +//*" + CI/") 

+ v^ (Va® - ffi) («7X" +//' + CI/") 
+ va (Vi® - 1^) (Ax' + 6/ + >' ) 

-/(VS® - (5) (Va23 - ?^), 

that is. 6c<I)(-) = (Va®-<ffi)(VaS-|^){/FZ + V'a(F+Z)-/l. 

What, however, is really required", is the value of 4>(+); to find this, we have 
6c4> (+) = 6c^ (-) + 26cir 

=(V^(ff-(E)(vai8-i^)(/Fz+va(F+z)+/} 

+ 26cir - 2/(Vaff - (K) (VaS - 1^), 

the second line of which is 

2 (Va® - «i!) (v^is - 1^) 1^ (vn + ®) (vas + 1^) -/| 
^2 (Va@-(SHVal8-i^) ^(^^^^^(^^^^)_^^^^^, 

= 2 (Va® - <!E)(>/ii9 - 1^) va^, 

1 It may be shown without diffioolty that the (-) ngn would imply that the sections toaching 2 = 0, x=0, 
and £=0, y=0, were sections toaching ;c=0 at the same point. By taking the (-) sign in each equation we 
should have the solution of the problem **to determine three sections of a surface of the second order, the two 
sections of each pair touching one of three given sections at the same point," which is not without interest ; 
the solution may be completed without any difficulty. 



114] steiner's extension of malfatti's problem. 67 

where 

and consequently 

iK5<i>(+)==(V^(aD-ffi)(VaS-f^){/Fz+va(F+z)+/+2tfVa}, 

a reduction, which on account of its peculiarity, I have thought right to work out in 
full. 

The condition of contact is 

4> (+) = VT'' = JL ( VS(^ - ffi) ( vai8 - 1^) vr+T^ ViTzi 

Hence finally, the condition in order that the sections 

(the former of which is a section touching £^ = 0, d; = 0, and the latter a section touching 
ir = 0, y = 0) may touch, is 

/Fz+va(F+z)+(/+2^va)-V6^vrTT^vnr^=o. 

The preceding researches show that the solution of Steiner's extension of Malfatti's 
problem depends on a system of equations, such as the system mentioned at the 
commencement of the following section. 

§5- 
Consider the sjmtem of equations 

a +i8 (F+z)+7 Yz j^h vnnr»vr+z» =0, 

«" + y8" (Z + F) + 7"ZF+ 8" Vr+X» VTTP = ; 

these equations may, it will be seen, be solved by quadratics only, when the coefficients 
satisfy the relations 

/3 yS' /3" 



7-a y-a' V'-«"' 

,y»_a» " 7'»-a^ 7"»-a"» 

9—2 



68 ANALYTICAL RESEARCHES CONNECTED WITH [ll4 

It may be remarked that these equations are satisfied by 

/8 = 0. /3' = 0, /3" = 0. 7 = 8, 7'=S', 7" = ^ 

or if we write 

a , o' a" 

- = — ^ -, = — m, -77 = — w, 

7 7 7 

the equations become by a simple reduction, 

7»-hZ*+2i FZ = P -1, 
Z*-h Z«+ 2mZZ = m» - 1, 
Z»+P-h2nZF=n» -1, 

which are equivalent to the equations discussed in my paper " On a system of Equations 
connected with Malfatti's Problem and on another Algebraical System," Cambridge and 
Dublin Mathematical Joumat, t. iv. [1849] pp. 270 — 275, [79]; the solution might have 
been effected by the direct method, which I shall here adopt, of eliminating any one 
of the variables between the two equations into which it enters, and combining the 
result with the third equation. 

Writing the second and third equations under the form 

^" + 5"z + cvr+z* = 0, 

the result of the elimination may be presented in the form 

which is most easily obtained by writing X=tan0 and operating with the symbol 
cos^^; but if the rationalized equations be represented by 

V + 2/Z + i;'Z» = and V' + 2/'Z + i/''Z» = 0, 

the form 

4 {W - /i'«) {\%" - /'») = {W + \'V - 2//i'7 

leads easily to the result in question. The values which enter are 

(7 = 8' VIT^, C" = S" VTTF'; 

whence, in the first place, by the equation connecting F, Z, 

C(7" = -?|l'{a + ^(F+Z)-h8FZ}. 



114] steinbr's extension of malpatti's problem. 69 

It is obviously convenient that A'A"-\-BB' should be symmetrical with respect to 
F and Z, and this will be the case if 

that is, if /8'(y'-a'0=/3"(7'-«'); 

or assuming that the equations are symmetrically related to the system, we have the 
first set of relations between the coefficients, relations which are satisfied by 

a = 7 + 20i8, a' = y + 20i8', a'' = 7" + 20/8", 

and the values of a, a\ a" will be considered henceforth as given by these conditions. 
We have 

il'^'' + 5'^'' - C^C^' = o'o'' + /8'/8'' + (7'/3'' + 7^/8' + 20/3'i8'O ( F + Z) + (^^^ 

H-^{a + ^(F-hZ)H-7FZ}. 

The quantities A'^-^-B^— C\ A"^ + B'^ — (7'» are quadratic functions of Z and Y respectively, 
and with proper relations between the coefficients, we may assume 

^A'^'^B^'-C'^){A'''''^B'''^G'^)^W{V^^-kl{a^fi{Y^Z)^-r^YZy 

in which 17 is a linear function of Y '\- Z and FZ, and h and U are constants. The 
first side must, in the first place, be symmetrical with respect to F and Z, or 

must be proportional to 

But since 

(«' + 7')/8'. (a" + 7")/9" 

are proportional to 



it is only necessary that 



should be proportional to 



y«-.a'«, 7"»-a"», 



^a + yi-S'*, /3"« + y'«-S"» 



y«-a'», 7"*-a"»; 



or since the equations are supposed symmetrically related to the system, we must have 
the second set of relations between the coefficients. Suppose 



then since 



y-a» = -4(7 + 0/8)0)8, &c.. 



70 ANALYTICAL RESEARCHES CONNECTED WITH [114 

we have 8» «/8« +7» -*8(y +^fi)fi 

and S, S\ S"' will be supposed henceforth to satisfy these equations. 
We have next 

which may be simplified by writing 

u - <& l-\-ii4> 

g as t- ^ w :s 2-— 

where ^ v are to be considered as given functions of a and <f>. These values give 

A'* +R*-Cr* =4(7' +4>$')^s {Z + ,i){Z + v), 
^"»+ £"«_ C"t = 4 (y + 0y3") ff'8{Y+it) (F+ 1/). 

Hence, putting for simplicity 

we have 

4(Z + /*)(Z+i;)(F+At)(F+v)=£7» + *[(a+/3(F+Z) + 7F^-S'(l + F»)(l+Z')]; 

and the two sides have next to be expressed in terms of T + Z and YZ. 

If for symmetry we write 

f = l. 7, = 7+Z, K^YZ, 
then 

and U' is now to be considered a linear function of f, 17, 2^. 

The condition that the first side of the equation may divide into factors, gives an 
equation for determining k ; since the condition is satisfied for A; = and k-<x>, the 

equation will be linear, and it is easily seen that the value \a k^-^iji — vf. In fact 

hence 

{2,*vf + 0* + v) 1; + 2f }« - t7» = ^^i^ {(af + /3i, + 7f )• - S* (f + ?)■}. 



114] steineb's extension of malfatti's problem. 71 

and we may assume 

2Mvf + 0* + .;)i7 + 2?-O-=^^{(af + ^i, + y?) + 8(f + ?)}, 
subject to its being shown that 

gives a constant value for A. The comparison of coefficients gives 



*~ a 

the first and third of these give 



{(A.l),-(A-i)4; 



4 (1 - ^v) = ^g" (a + ^) (7 - «), 



which will be identical with the second, if 

2(1 -Mv) ^ ^ ^ gA 

which follows at once from the equation 



Forming next the two equations 



ns 


1/ = 










A + 


1 

a' 




2 


(M + 


•')«, 




"(m- 


-')/3 




A- 


1 
A' 


"(m 


2 

-«'))3 


Km + 


i')7- 


-2/9}, 



these will be equivalent to a single equation if 

(m + !/)• 8»= {0* + »')7 - 2/3{» + (/* - 1/)* /8», 
that is, if 

0* + !/)• S» = 0* + 1/)" (/S* + 7») - 4 (/i + v) i87 - 4 (/*•; - 1) /S* ; 



72 ANALYTICAL RESEARCHES CONNECTED WITH [114 

or finally, if 

which is in fact the case. 



Writing ihe equations for 



A 1 A 1 



in the form 



-^ + T = ;^ \"^-^"- x = 7 r5-(7- 2w, 

A (ji — vjps A {fi-vjPs^' 



and substituting in 



we have 



U='^\[^-^^(<'i+N+'iK)-[^+^»{^+K)\, 



^ ((-/3 + 2«7 + 2^)f + (7-2«/3)i7 + (-/9+2»7 + 4«^/8)?}; 



and consequently, multiplying by 
we have 

or collecting the different terms which enter into the equation 
the result is 

- 1 V(7' + 2^/3')(7" + 20y8")/8'/8"l {(- /3 + 2*y + 2^) f +(7 - 2»^) 17 + (- /8 + 2*/ + 4«^/9)r} = 0. 

which, combined with the first equation written under the form 

(of + ^, + yO* - «• [(f - ry + 17»] = 0, 
determines the ratios of ^, 17, ^ that is, the values oi T + Z and YZ. 



114] STiaNEB's EXTENSION OP MALPATTl's PROBLEM. 73 

§6- 
The system of equations 

(/+ 20 V^) + V|l (F+ Z ) +/YZ - \/bc ^/T+Y' 'JT+Zi = 0. 

(g + 26 vIB) + VS(^ + X) +gZX - Vw \/l + Z* Vf+Z* = 0, 

(A + 2^ V^) + V® (Z + F) + AXF - V^ Vr+X« VlTT' = 0, 
where 

on which depends the solution of Steiner's extension of Malfatti's problem, is at once 
seen to belong to the class of equations treated of in the preceding section, and 
we have ^ = 0, « = 0. The equations at the conclusion of the preceding section 
become 

-a[(/+2^Vgr)f + Va, +/r] -^-J(g+e VS) (h+e V®) Vig® {('^-20f)^-fy + V|i?}=o. 

{(/+ 2^ va) I + v^, +/f}« - be {(f - (0* + »?'} = 0, 

which may also be written 

(VS® + Jf) (f + 5) + (- a Va + «7 V® + A ^18 + 25 Vl8®) (ij + 2^f ) 

- ^ -Jig + e VlJ) (A + 5 V®) Vfi®! ((VI - 25/) f -/i; + '/^d = 0, 



{/«+D + ^('? + 25f)}'-^Kf-0' + i?'} = 0. 



Hence observing that 



^+5Vii = ^(Vi8® + JF)(Vai8 + |^); A + dV® = -^(VJ8®+iF)(Va® + ffi); 

-oVa+AVS+5'V®+25VS®=5(VS®+jp), 

and putting for a moment 

X = ^ ^/(V^ + (5) (Vaa8 + 1^) V3B© , 

and therefore 



V(^+ 5 VS)(A + 5 V«D) Vi8aD=(VJ8® + iF)x, 
a n. 10 



74 ANALYTICAL RESEARCHES CONNECTED WITH [114 

the first equation divides by (^^^8CD + Jp), and the result is 

Also, by an easy transformation, the second equation becomes 
or putting 

f+? + ^ (i? + 2^f ) = e, 



the equations become 



hence eliminating ^, 



e - 2x,o = 0, 



or observing that 



1 + ^ = ;^. (^^® + iF) (v^^a + ®) (>^ai8 + ?^), 



and reducing, we obtain 



*=.^-=^ 



also B = 2X<I> gives 









Suppose 



then substituting 



V3iar+ jF = '*' V3S®- iF = «,. •■• «», = Ka, 

P/r, \ Va + a/ 



114] STEINEE's extension of MALFATTfs PROBLEM. 75 

that is, 

Pfli 



P.7/ V Va + a/ 



these may be written 

L'^ + if 17 + i\r'c= 0, 

where 

P,7/ V Vo+a/ 
or since f, ly, ? are equal to 1, T+Z, YZ respectively, 

1: Y + Z: YZ= MIT - MN : NL' - ITL : LM - L'M 

^_46c^)/ V2V^ A / _ ^U^J+3(/+,Va). 
^-% ^ V/8,X >^ ^ Va + a/ V/8,7, 

Also 

whence 

y ^^^ _ 2V2V^Tv^/l _ V2^^^^N / _ V^N _ 

^ ^ V)8,x A Va + a/ 

2V2 V^T^, ^^ /^tf , ^2 Va + a/ \ / V;; N 

^ V "^ v;^, Jr va+v ' 

and by forming the analogous expressions for Z-\-X and ZX, X+F and XY, the 
values of X, F, Z may be determined. But the equations in question simplify them- 
selves in a remarkable manner by the notation before alluded to. 

10—2 



76 ANALYTICAL RESEARCHES CONNECTED WITH [114 

Suppose 

these values give 

iJ-^ =-f»-g'-h» + 2J>, 

K* =-f«-g'-h« + 2g»h» + 2h»f+2fV-^^'. 

Applying these results to the preceding formulae and forming for that purpose 
the equations 

» /s / la- A 1. ^2 Va + o" J* Vo, f 

2 V2 Va + a, V^,7, = 4gh, 73 =-7s^. , — ^ = 7. 

V^,7/ Vagh Vo + a, «/ 

ghA-^ + ^ ITf = ( J« - gh) (f» - (g - h)*) - 2gh (g - hy. 

we have 

if(F+^ + 2Z^ = 4(J*-gh)(^l -^), 

iTTZ + Z« = {(J' - gh) (f» - (g - h)*) - 2gh (g - h)«} (1 - ^) ; 

the former of which, combined with the similar equations for Z-\-X and -T + F, gives 
for Z, F, Z the values to be presently stated, and these values will of course verify 
the second equation and the corresponding equations for ZX and XY. 

Recapitulating the preceding notation, if j? = 0, y = 0, z^O are the equations of the 
given sections, t(;=»0 the equation of the polar plane of their point of intersection 
with respect to the sur&ce, 

CM5" + 6y* 4- c-«* + 2^2? 4- 2gr^ + 2fcry +/)«;» = 
the equation of the sur&ce, ^, 38, ({D, $, <!Er, |^, iT as usual, and 



114] steikeb's extension of malfatti's problem. 77 

then the equations of the required sections are 

{ax + hy + gz)X'^y V® + ^ ^^ + w V - op V 1 + Z« = 0, 

X V® + y Va + (gx +/y + cz) Z+w^ - cpVl + Z* = 0, 
where X, F, ^ are to be determined by the following equations, 

(/+ 25>/a)+ va (Y+z)-hfYz - \/6c vrrF^viT^ = o, 
(y + 2^ VS) +vs(.^+z)+^^z-VcaV rr^ vi+z« = o, 

(A + 25 V®)+ V^(Z + F) + AZF- \/^ Vl+Z«Vrn^ = : 
and the solution of which, putting 

f=4'a>/vi5«^rS g=^©^/^^®a-ffi, h=^®>/vai3-?^, j^j~2<fmm. 

is given by the equations 

/irz = ^+( -.f+g + h)«-2(-f+g + h)j, 

ifF=?^ + ( f-g + h)>-2( f-g + h)J, 

ZZ = ?^^+( f + g-h)>-2( f+g-h)/. C) 

Instead of the direct but very tedious process by which these values of A", F, Z 
have been obtained, we may substitute the following d posteriori verification. 

We have 

K*(l+X*) =4(-f+g + h)'J'(l+^)(l-j)(l-J). 

2f,VlTF»VrT^ = 4 (f -(jf-h )) /• (l - j) \/l - $y/l - J, • 

/r»(l + FZ) = * (l - j) {('^' -«*»> i^-(g-^y) - 2gh(g - h)",. 

^(F+Z)-2f»- 2g«- 2h« + 4/» = 4 (l -^) (J'-gh). 
Putting also 

^ It IB periiapB worth noticing that the value of the quantity X previously made use of, 



78 



we have 



ANALYTICAL RESEARCHES CONNECTED WITH 



[114 



(f. _ g. _ h« + ?^') IT* (1 + FZ) 

= 4(l-J)|(p-(g-h)')[(J'-gh)(f«-(g-h)«)-2gh(g-h)«-2gh^=^^5^] 



4g' h'(g-h)'(J»-g h) 



}■ 



hnj*-gh)\ 



J' 



K' {K(Y+Z) - 2f«- 2g'- 2h' + 4/'} 

= 4(i-;)j{f-(g-hy)[(j»-gh)((g+h).-f*-^?:)l-*«^- 

Also, since 

(f._(g.h).) + ((g+h).-f^-*^') = 4gh(:^>. 

we have 

(p -^ -h' +^^\ K' (1 + YZ) + K* {K (Y + Z) -it* -2^ -2h*} 

= 4(l-;J) (f«-(g-h).)2ghJ« (l -^.)(l- ^,). 
and the values obtained above give also 

2gh \/l - ^\A-J. ^' ^ + ^' ^^1+^' 

=4(i-^)(f«-(g-h)02ghj«(i-f,)(i-^;). 

which shows that the relation between Y and Z is verified by the assumed values of 
these quantities, and the other two equations are of course also verified. The solution 
of the problem will be rendered more complete if the equations of the required sections 
and of the auxiliary sections made use of in the geometrical construction are expressed 
in terms of f, g, h, J. 



§7. 

First, to substitute in the equations of the required sections or resultors. 
the first equation in the form 



Writing 



K' 



the coeflBcient of x will be 



|aZa? + (AZ + V(!D)y + (grZ + V33)-^ + V-apVl + Z«w=0^ 



-^ ( 



^-j^l^'+i-f+s+^y-^i'-f-^g-^^)]^ 



114] 



steiner's extension of malfatti's problem. 



79 



or, as it is convenient to write it. 



1 + 



^)r(-'.».h,4(i-^){7p^^-f...h-^. 



The coeflBcient of y is 



^{(-f.-g.^-.+^^X^'^*-'^*-'"^-"'-'*^^'')) 



2Vi3 



f 4 _ g4 _ h« + 2g»h' + 2hf + 2f g» - ^^^ 



or, after all reductions, 



i} 4)«-'-^-«^ (> -« {.-F^,*'-«-"-^m^h 



and similarly the coefficient of 2^ is 



^|(-f.g--H-..^-e-)(?&t.(-,.,.h).-2^(-f.g^h,) 



_ f « _ g4 _ h« + 2g»h» + 2h»f + 2Pg' - ^^ j 



or, after all reductions. 



(,4)„.,.,..,^(,.h)|_^||^.,,,,.,,^2£^.b.,l 



and the coefficient of u; is 



(i + J)f(-f+g+h)2Vzyi-iy^i-&yi-^V-jt,. 



Hence, forming the equation of the resultor in question, and by means of it those 
of the other resultors, the equations of the resultors are 



, _2fgh_ 
U(-fVg + h) 



- f + g 4 h - 2/ 



)U-'^' 



+ U(-f+g + h) + ^ g + h + 2J 2^^ j^\\ jjy 
I -2fgh .,, . orf*-g» + h»\ h /, h\ 



80 ANALYTICAL RESEARCHES CONNECTED WITH [114 



+ 2 VF/^r^ /y/lTK y^T^ V^«, = 
values which might be somewhat simplified by writing |, t), ^, <o instead of 

U'-'i)'- U-i>' U'->- V^^/'^V'"^^- 

and it may be also remarked, that the coefficients as well of these formulae as of those 
which follow may be elegantly expressed in terms of the parts of a triangle having 
f, g, h for its sides. 

The equations of the separators are found by taking the differences two and two 
of the equations of the resultors (this requires to be verified d posteriori) \ thus sub- 
tracting the third equation firom the second the result contains a constant fector, 

j(f.-(g-h>')gh {^^g"^' -^ (f - (g - h)*) ((g+ by - p)}. 

equivalent to 

J(P-(g-h)^)ghV ^ V J^ J) (f«-(g-hy)gh 

Rejecting the factor in question and forming the analogous two equations, the equa- 
tions of the separators are 



^ (aa' + ...:? S*) = (aa + f^/8 + (Syy, 



114] steiner's extension of malfatti's pkoblem. 81 

and from the mode of formation of these equations it is obvious that the separators 
have a line in common. 

The equations of the determinators being x = 0, y = 0, e^Q, the equations of the 
tactors are 

v'S«-V^y = 0, Vcf a; - V^« = 0, Vay- V3©a;=0; 

and if ax + fiy + yz + Bw = be the equation of the tactor touching 

01 = 0, V®a!-v'a« = and Viiy-VSa! = 0, 

the conditions of contact are 

K 
P 

2 V15B ( VIS - 1^) (aa" + . . . - 8*) = |( VISB - 1^) (a Va -/3 VS) + 7 (® VS - 1^ V^)l* , 

2 VI® ( va® - ffi) (aa* + . . . - s*) = Ws® - (5) (a vfi - 7 V®) + ^ (?^ v® - jf va)l* . 

whence 

;;^V2Va3B(VaS-|^)(aa + |^/3 + €J7) = 

(Vgoi - ^) vaa - (VIS - ^) vs/s + (ffi vs - jf va)7, 
^V2Va®(va®-ffi)(aa+|^i8+ffi7) = 

(Va® - ffi) Vgfa + (1^ V® - Jf VI) /3 - (Vie - <!Ir) 7, 
and putting for a moment 

/i = vae - (5 - V2"viiaD(Val-€G), 



V = Vi® - 1^ - 72 vas (Vai8 - 1^); 

after some reductions, and observing that the ratios only of the quantities a, fi, y, S 
are material, we obtain 

K 



c. n. 



va" 

11 



82 ANALYTICAL RBSBARCHB8 CONNECTED WITH [ll4 

and it is easily seen also that the coordinates of the point of contact are 



also 



« = 0, y = v, z^fi, t^ = - - -^- 5 



'•- ^('-5). '=-^('-5)- 



Hence substituting and introducing throughout the quantities f, g, h, «/, also forming 
the analogous equations, the equations of the tactors are 

jp(-f.+g.+h.)+(g+h)j(f.-(g-h)--?^)j^. 



+ 2>JK V gh (l - 5) (l - j) (f - (g-h)*) 'J~PW = 0, 
+ 2V^ y hf (l - J) (l - j) (g" - (h - f )') -J"-^ = 0, 

+ |h. (f. + g. - h«) + (f + g) / (h» - (f - g)» - ^ijj-*)! -^ ^ 

^•JKsJigil - j) (l - f) (h'-(f-g)') ^^^^> = 0- 



+ 



114] steiner's extension op malfatti's problem. 83 

It is obvious, from the equations, that each separator passes through the point of 
contact of a tactor and determinator, it consequently only remains to be shown that 
each separator touches two tactors. Consider the tactor which has been represented 
by aa; + /8y -f 7-8r 4- Sw = 0, the unreduced values of the coefficients give 



^aa«+...|S» = ;;^(aa + f^5 + (E7) = iri 



Represent for a moment the separator 



h{^-7)'-U-?)y-'-^'U-^>"' 



hy lx + my ■hnz-h8W=: 0. Then putting ^P + ... — s» = Q', since 



= ^{,(i4)-,(i-^)(i-5)-(r-g)(i-^)(i-i)} 



the condition of contact becomes 



D=-r|-(f-g)'+h(f+g)-?^|; 



J 
or, forming the value of □' and subetituting, 

-.(.-^(r-,„(i_5)(,4)-.(,.^>,(,.5)(,_5) 

which may be verified without difficulty, and thus the construction for the resultors 
is shown to be true. 

11—2 



84 



ANALYTICAL RESEARCHES CONNECTED WITH 



[114 



§8 

Several of the formulae of the preceding sections of this memoir apply to any 
number of variables. Consider the surface (Le. hypersurface) 

aai^ + bf + C2^+ 2fyz + 2gza) + 2hxy ... +pe* = 0, 
and the section (i.e. hypersection) 



where 



(aX + A/i +5fv ...) a? + {h\ + bii +fv ...) y + (ff^ +/a* H- ci' ••.) ^ ••• + ^ ""^ ^^ = 0, 



V* = a\* + bfi* + cv" + 2//X1/ H- 2gv\ + 2AV ... - K, 



the condition of contact with any other section represented by a similar equation is 

a\V + bfJLfi' + cw' +f(jiv + /jfp)-hg (vXf + v\) + h (Kf/ + \» ... ±K = W, 

where K is the determinant formed with the coeflScients a, 6, c, /, g, A, ... And con- 
sequently, by establishing all or any of the equations \ = \/^, /a = >/18, v = VCD, ... 
we have the condition in order that the section in question may touch all or the 
corresponding sections of the sections a? = 0, y = 0, z = Of ... 



Let tt be the number of the variables x, y, z... , then K^"^ =. 



G :ff (S^ 



also 



Z«-» {(a\ + hfM + gv ...) x + (h\ + bfi +fa ...) y + {9>^ -^ff^ + op ...) z ...] 



X 


y 


2 


X a 


1^ 


ffi 


A* ?^ 


IS 


JF 


I* Ci 


iF 


® 



whence also 








Z«-*(V» + ^ = - 


X /» 


V ... 


or /ir"-*v> = - 




X a 1^ 


e 






^ ?^ 53 


JF 






1 " ffi iF 

1 • 


® 





1 


X 


/* 


V 


X 


^ 


?^ 


® 


/* 


n 


19 


iF 


» 


€r 


iF 


€ 



114] 



STEINBBS EXTENSION OF MALFATTIS PBOBLEM. 



85 



and the equation of the section in question becomes 



X 


y 


z ... 


+ ir*«->v-j>v- 


\ & 


n 


& 




t* n 


i3 


JF 




V <& 

• 
• 


iF 


e 





1 \ 



iF 



also the condition of contact with the corresponding section is 



Tl 


X 


/* 


V 


V 


» 


?^ 


dr 




1^ 


13 


iF 


l/ 


eEr 


iF 


® 



V- 


1 X 


/* 


V ... 




X a 


1^ 


<!1t 




M n 


13 


iF 




• 
• 

• 


iF 


® 



V ... 

iF 



^ 



« = 0, 



V- 


1 X' 


/*' 


y'... 1 




X' a 


^ 


dr 




/*' » 


as 


iF 




v' ® 

• 
• 
• 


iF 


e 



In particular the equation of the sections which touches all the sections x^O, y 
«=0, ..., is 



= 0, 



X y z 

V® <S jp CD 



+ 



if jii-i V - p yn 



1 Via vs v®... U=o. 

va a ?^ ffi 

v^ f^ 18 iF 

V^ (ffir iF ffi 



Again, the equations of the section touching y = 0, ^ = 0,... and the sections touching 
«sO, ^ = 0, ... are 

> = 0, 





tc 


y 


z 


X 


a 


?^ 


dr 


v13 


?^ 


33 


iF 


• 
• 
• 


df 


iF 


e 




a; 


y 


2 


va 


a 


?^ 


dr 


^ 


?^ 


13 


iF 


V® 


df 


iF 


® 



+z»«-w-j3 7- 


1 

X 

V® 

• 
• 
• 


X Vi3 V®... 

a ?^ dr 
1^ 55 iF 

di JF ® 


+ Jfi«-W-p>/^ 


1 

V® 

• 

• 
• 


va" /* ^ - 
a ?^ dr 

1^ 3D iF 

ffi JF ® 



1< = o, 



86 



ANALYTICAL RESEABCHE8, &C. 



[114 



and the condition of contact of these two sections is 



+ 1 


X 


Vi3 


m 


sa 


1^ 


/* 


m 


13 


V® 


& 


iF 



V<ZD... 

JF 



= V_" 



1 


X 


^23 V®... 


\ 


^ 


1^ cs 


V39 


1^ 


38 iF 


V@ 


<!1t 


iF ® 



y- 



1 va 



18 iF 

iF ® 



It would seem from the appearance of these equations that there should be some 
simpler method of obtaining the solution than the method employed in the previous 
part of this memoir. 



115] 



87 



115. 



NOTE ON THE PORISM OF THE IN-AND-CIRCUMSCRIBED 

POLYGON. 



[From the Philosophical Magazine, voL VL (1853), pp. 99 — 102.] 

The equation of a conic passing through the points of intersection of the conies 

17=0, F=0 
is of the form 

wU+V=0, 

where i(; is an arbitrary parameter. Suppose that the conic touches a given line, we 
have for the determination o( w a, quadratic equation, the roots of which may be 
considered as parameters for determining the line in question. Let one of the values 
of w he considered as equal to a given constant k, the line is always a tangent to the 
conic 

kU+V = 0; 

and taking w^^p for the other value o{ w, p is a parameter determining the particular 
tangent, or, what is the same thing, the point of contact of this tangent. 

Suppose the tangent meets the conic U = (which is of course the conic corre- 
sponding to w = co)m the points P, P^, and let 0, oo be the parameters of the point 
P, and ff, 00 the parameters of the point P'. It follows from my "Note on the 

Geometrical representation of the Integral Jdx-i- V(aj + a) (a? + 6) (x + c)," [113] (*) and 
from the theory of invariants, that if nf represent the "Discriminant" of ^U + V 

^ I take the opportonity of oorreoting an obvious error in the note in question, viz. a'+&'+c*-2&c-2eii-2a5 
is throughout written instead of (what the expression should be) 6*c'+c'a*+a*&'-2a'&e-2&>ea-2c^. [This 
coiraotion is made, ante p. 55.] 



88 NOTE ON THE PORISM OF THE IN-AND-CIRCUMSCBIBED POLYGON. [ll5 

(I now use the term discriminant in the same sense in which determinant is sometimes 
used, viz. the discriminant of a quadratic function cue* + iy* + c-g^' + %fyz + 2gzx + 2hxy 
or (a, 6, c, /, g, h) {x, y, zy, is the determinant k = abc — of* — 65^ — ch* + 2fgh), and if 



•^ 00 



df 



then the following theorem is true, viz. 

"If {0y 00), (ffy 00) are the parameters of the points P, P' in which the conic 
U=0 is intersected by the tangent, the parameter of which is p, of the conic 
kU+V=0, then the equations 

n^ ^np-Uk, 

n^=np+nA:, 

determine the parameters 6, ff of the points in question." And again, — 
"If the variable parameters ^, ff are connected by the equation 

n^-n^=2nit, 

then the line Ff will be a tangent to the conic fcCr+F=0." Whence, also, — 

" If the sides of a triangle inscribed in the conic 17= touch the conies 

k I7+F=0, 
it'I7+F=0, 

rr7+F=o, 

then the equation 

nifc+nifc'+nr=o 

must hold good between the parameters fc, k\ VT 

And, conversely, when this equation holds good, there are an infinite number of 
triangles inscribed in the conic [7=0, and the sides of which touch the three conies ; 
and similarly for a polygon of any number of sidea 

The algebraical equivalent of the transcendental equation last written down is 



1, k , Vn* 
1, V, VqF 



= 0; 



1, y\ VqF 

let it be required to find what this becomes when k^V ^kf' ^0^ we have 



115] NOTE ON THE P0RI8M OF THE IN-AND-CIRCUMSCRIBED POLYGON. 89 

and substituting these values, the determinant divides by 

1, k\ ifc'» 

1, r, r« 

the quotient being composed of the constant term C, and terms multiplied by k, k\ k" ; 
writing, therefore, fc = A/ = fc" = 0, we have (7=0 for the condition that there may be 
inscribed in the conic Z7 = an infinity of triangles circumscribed about the conic 

F=0; C is of course the coeflBcient of f" in Vnf> i-e. in the square root of the 
discriminant of f 17"+ F; and since precisely the same reasoning applies to a polygon 
of any number of sides, — 

Theorem. The condition that there may be inscribed in the conic CT^O an 
infinity of 7»-gons circumscribed about the conic F = 0, is that the coefficient of f*"' in 
the development in ascending powers of f of the square root of the discriminant of 
f[7+ F vanishes. [This and the theorem p. 90 are erroneous, see po»^, 116]. 

It is perhaps worth noticing that w = 2, i. e. the case where the polygon degene- 
rates into two coincident chords, is a case of exception. This is easily explained. 

In particular, the condition that there may be in the conic ^ 

flwr»+6y*-|-cz'=0 

an infinity of »-gons circumscribed about the conic 

ic*+ y'+ -^' = 0, 
is that the coeflScient of 1^^ in the development in ascending powers of f of 

\/(l + a^)(l+6^)(l-hcf) 
vanishes ; or, developing each factor, the coefficient of f**~* in 

(l + iaf-Ja«f»-hT'ffa'?-?fia'^'+&c.)(l + i6f-&c.)(l + ic^-&c.) 
vanishea 

Thus, for a triangle this condition is 

a^ + 6» + c» - 26c - 2ca - 2a6 = ; 
for a quadrangle it is 

a» + 6* + c* - 6c«- 6»c - ca» - c^a - a6» - a'6 + 2a6c = 0, 

which may also be written 

(6 + c - a) (c + a — 6) (a + 6 - c) = ; 
and similarly for a pentagon, &c. 

' I have, in order to present this result in the simplest form, purposely used a notation different from 
^t of the note above referred to, the quantities ax>+&y*+c;e' and o^-h-y'^-^'Z^ being, in fact, interchanged. 

c. n. 12 



90 NOTB ON THE PORISM OF THE IN-AND-OIRCUMSCRIBED POLYGON. [llo 

Suppose the conies reduce themselves to circles, or write 

i2 is of course the radius of the circumscribed circle, r the radius of the inscribed 
circle, and a the distance between the centres. Then 

f[7+F=(f+l, f+1, -fi?^i- + a«, 0, -a, 0)(x, y, 1)«, 
and the discriminant is therefore 

Hence, 

Theorem. The condition that there may be inscribed in the circle a^ + y* - ii*^ = 
an infinity of n-gons circumscribed about the circle (j?— a)*+ y* — r* = 0, is that the 
coefficient of f"""* in the development in ascending powers of f of 



may vanish. 
Now 

(A +Bf + 6'?)* = VJ |l + i5 1 + (i^C - i5«) f. + ...} . 

or the quantity to be considered is the coefficient of f^^ in 

(1 +*f-4p-) {i + ifi|+ (i^c- 45«)|;+ ...|,. 

where, of course, 

In particular, in the case of a triangle we have, equating to zero the coefficient 
of p, 

or substituting the values of A, B, C, 

(a» - i?)« - 4r»i? = 0, 

that is 

(a» -i?+ 2iJr)(a»-i?- 2iJ7) = ; 

the factor which corresponds to the proper geometrical solution of the question is 

a»-i? + 2iJr=0, 

Euler's well-known relation between the radii of the circles inscribed and circumscribed 
in and about a triangle, and the distance between the centres. I shall not now discuss 
the meaning of the other factor, or attempt to verify the formulae which have been 
given by Fuss, Steiner and Richelot, for the case of a polygon of 4, 5, 6, 7, 8, 9, 12, 
and 16 sides. See Steiner, CreUe, t. n. [1827] p. 289, Jacobi, t. in. [1828] p. 376 : 
Richelot, t. v. [1830] p. 250; and t. xxxviii. [1849] p. 353. 

2 Stone Buildings, July 9, 1853. 



116] 



91 



116. 



COREECTION OF TWO THEOREMS RELATING TO THE PORISM 

OF THE IN-AND-CIRCUMSCRIBED POLYGON. 



[From the Philosophical Magazine, vol. VL (1853), pp. 376 — 377 ] 



The two theorems in my " Note on the Porism of the in-and-circumscribed Polygon " 
(see August Number), [115], are erroneous, the mistake arising from my having in- 
advertently assumed a wrong formulae for the addition of elliptic integrals. The first 
of the two theorems (which, in fact, includes the other as a particular case) should be as 
follows : — 

Theobem. The condition that there may be inscribed in the conic 17 = an 
infinity of n-gons circumscribed about the conic F=0, depends upon the development 
in ascending powers of f of the square root of the discriminant of fZ7+ F; viz. if 
this square root be 

then for n = 3, 5, 7, &c. respectively, the conditions are 



|C|=0, 



C, D 

D, E 



= 0, 



C, D, E 

D, E. F 

E, F, Q 

*nd for n = 4, 6, 8, &c respectively, the conditions are 



= 0, &C. ; 



ID 1=0, 



D, E 

E, F 



= 0, 



= 0, &c. 



D, E. F 
D, F, 
F, G, H 

^6 examples require no correction; since for the triangle and the quadrilateral 
'^spectively, the conditions are (as in the erroneous theorem) (7 = 0, Z) = 0. 

12—2 



92 CORRECTION OF TWO THEOREMS RELATING TO THE PORISM, &C. [llG 

The second theorem gives the condition in the case where the conies are replaced 
by the circles ^r* + y* — i? = and (a? — a)* + y' — ^ = 0, the discriminant being in this case 

- (1 + f ) {r» + f (r« + i? - aO + f^iP}. 

As a very simple example, suppose that the circles are concentric, or assume 
a = ; the square root of the discriminant is here 

(l+f)\/r» + i?f; 
and putting for shortness -^ = «» we may write 

il+5f+...=(l+f)VrTif, 
that is, ^=1, fi = ia + l. C = -ia» + ia», 2) = tV«'-*«"> ^ = -TfF«* + TV«'> &c. ; 
thus in the case of the pentagon, 

C^-D» = j^a*{(a-4)(5a-8)-4(a-2)«l 
= tAi «*(«*- 12a + 16), 

and the required condition therefore is 

a»-12a + 16 = 0. 

It is clear that, in the case in question, 

-^ = cos36° = i(^5 + l), 

that is, - = V5-1, or (12 + r)«-5r«= 0, 

viz. (Va + 1)" - 5 = 0, or a + 2 Va- 4 = 0, 

the rational form of which is 

(^- 12a + 16 = 0, 

and we have thus a verification of the theorem for this particular case. 

2 Stone Buildings, Oct. 10, 1853. 



117] 



93 



117. 



NOTE ON THE INTEGRAL Idx^Jim-^x) (x + a){x + b){x + c). 

[From the Philosophical Magazine, vol. vi. (1853), pp. 103 — 105.] 

If in the formubB of my " Note on the Porism of the in-and-circumscribed Polygon/' 
[115], it is assumed that 

and if a new parameter cd connected with the paramieter w by the equation 

com 

w= 

m — G) 

be made use of instead of w, then 

and thus the equation wU+V=Oy viz. the equation 

G)(d5» + y" + -2^) + aa:' + 6y« + C2» = 0, 

is precisely of the same form as that considered in my "Note on the Geometrical 
Representation of the Integral Idx -=- V(a: + «)(« + 6) {x + c)," [113.] Moreover, introducing 
instead of f a quantity 17, such that 



then 

VDf "Jim-ri) (a + 17) (6 + 17) (c + 17)' 



94 NOTE ON THE INTEGRAL j dx-i- J{m-'X) (x + o) (x + b) (x + c). [ll7 

Also f = 00 gives 17 = m, the integral to be considered is therefore 

11,17=1 . -- — ; 

Jm v(m-i7)(a + i7)(D + i7)(c + i7) 

i.e. if in the paper last referred to the parameter 00 had been throughout replaced 
by the parameter m, the integral 

drj 



ni7=f -j= 



V(a 4- 17) (6 + 17) (c + 17) 

would have had to be replaced by the integral 11,17. It is, I think, worth while to 
reproduce for this more general case a portion of the investigations of the paper in 
question, for the sake of exhibiting the rational and integral form of the algebraical 
equation corresponding to the transcendental equation ±Tl,k±TI,p ±11,0=^0, Consider 
the point f, 17, f on the conic m(j:' + y' + ^) + aaj"-f fty" + C£i" = 0, the equation of the 
tangent at thife point is 

(m + a) f a? + (m + 6) i7y + (m + c) 5? = ; 

and if ^ be the other parameter of this line, then the line touches 

^ (a;» -f y " + ^) + OA-" + 6y> + c^ = ; 
or we have 

d + a "^ ^ + 6 ^ + c ~ ' 

and combining this with 

(m + a)f" + (m + 6)i7« + (m + c)f = 0, 
we have 

f : 17 : g'= ^ b "C 'Ja-^-d "Jb -k-m ^c+m 

: ^{c-'a)'Jb + 0^c -{•m'Ja + m 

: V(a '-b)'^C'\-0^a + m^b'k-m 

for the coordinates of the point P. Substituting these for x, y, z in the equation of 
the line PP" (the parameters of which are p, k\ viz. in 



a?V6--c v^(a4- A;)(a+p) + yVc-aV(6 + A;)(6+p) + 2:Va-6Vc + fcVc+;)=0, 
we have 



'Ja + m V6 + 



m 



vc + m 



117] NOTE ON THE INTEGRAL I dx-^ J{m-'X) (x + o) (x + b) (x + c). 95 

which is to be replaced by 

(a+p)(a-h^)(a + g)^ 

6 + m 
These equations give, omitting the common factor (a + wi) (6 + m) (c + m), 

+ {abc {kd-\-0p+ kp) -\-pk6 {bc-{-ca + a6)j, 

-{■m[''aJbc -pkO-^ (bc + ca-k-ab) (/> + *+ 5) + (*^ + ^[p +;>*)(« + 6 + c)} 
+ {oftc (p + ^* + ^) - pk0 (a + 6 + c)} , 

+ m {{be -{-ca-^- ab) - {k6 + 6!p +#)} 
+ ahc+pkO; 

and substituting in 4X*. ft' — (2X/a)* = 0, we have the relation required. To verify that 

the equation so obtained is in fact the algebraical equivalent of the transcendental 

equation, it is only necessary to remark, that the values of \\ /i' are unaltered, and 

that of \fi only changes its sign when a, 6, c, m and />, k, 0^ —m are interchanged ; 

and so this change will not affect the equation obtained by substituting in the equation 

4X' . ^' — (2X/a)' = 0. Hence precisely the same equation would be obtained by eliminating 

L, M from 

(ifc + a) (ifc + 6) (ifc + c) = (Z + Jfit)» (m - A), 

(p + a)(p + 6)Cp + c) = (Z + ifi>)*(m-p), 

(^ + a)(5 + 6)(d+c) = (Z + Jftf)«(d-j9); 

or, putting (Z + Mk) (m— k)='a + ^k + ^h?, by eliminating a, ^, 7 from 

(m - k){k + a) (k + 6) (i + c) = (a + /3A + yk'Y, 
(m-p)(p + a)(/> + 6)(/> + c) = (a + /3p + 7p«)«, 

(m-5)(5+a)(5+6)((9+c) = (a + /9^ + 7^)^ 

= (a + ^m + 7m')«, 

which by Abel's theorem show that p, i, ^ are connected by the transcendental equation 
above mentioned. 

2 Stone BuildingSy July 9, 1853. 



96 



[118 



118. 



ON THE , HAEMONIC RELATION OF TWO LINES OR TWO 

POINTS. 



[From the Philosophical Magaziney vol. VI. (1863), pp. 103 — 107.] 

The "harmonic relation of a point and line with respect to a triangle'* is well 
known and understood ^ ; but the analogous relation between two lines with respect to 
a quadrilateral, or between two points with respect to a quadrangle, is not, I think, 
sufficiently singled out from the mass of geometrical theorems so as to be recognized 
when implicitly occurring in the course of an investigation. The relation in question, 
or some particular case of it, is of frequent occurrence in the Traits des ProprOtis 
Projectives, [Paris, 1822], and is, in fact, there substantially demonstrated (see No. 163); 
and an explicit statement of the theorem is given by M. Steiner, Lehrsatze 24 and 25, 
Crelle, t. xiil [1835] p, 212 (a demonstration is given, t. xix. [1839] p. 227). The 
theorem containing the relation in question may be thus stated. 

Theorem of the harmonic relation of two lines with respect to a qvadrUaieral. " If 
on each of the three diagonals of a quadriUiteral there be taken two points harmonically 
related with respect to the angles upon this diagonal, then if three of the points lie 
in a liney the other three points will also lie in a line^* — the two lines are said to 
be harmonically related with respect to the quadrilateral. 

It may be as well to exhibit this relation in a somewhat different form. The 
three diagonals of the quadrilateral form a triangle, the sides of which contain the 
six angles of the quadrilateral; and considering three only of these six angles (one 
angle on each side), these three angles are points which either lie in a line, or else 

^ The relation to which I refer is contained in the theorem, *'If on each side of a triangU there be 
taken two points harmonicaUy related with respect to the angles on this side, then if three of these points 
lie in a line, the lines joining the other three points with the opposite angles of the triangle meet in a 
point,** — the line and point are said to be harmonically related with respect to the triangle. 



118] ON THE HARMONIC RELATION OF TWO LINES OR TWO POINTS. 97 

are such that the lines joining them with the opposite angles of the triangle meet in 
a point. Each of these points is, with respect to the involution formed by the two 
angles of the triangle, and the two points harmonically related thereto, a double point; 
and we have thus the following theorem of the harmonic relation of two lines to 
a triangle and line, or else to a triangle and point. 

Theorem. '' If on the sides of a triangle there be taken three points, which either 
lie in a line, or else are such that the lines joining them with the opposite angles 
of a triangle meet in a point; and if on each side of the triangle there be taken 
two points, forming with the two angles on the same side an involution having the 
first-mentioned point on the same side for a double point ; then if three of the six 
points lie in a line, the other three of the six points will also lie in a line," — the 
two lines are said to be harmonically related to the triangle and line, or (as the case 
may be) to the triangle and point. 

The theorems with respect to the harmonic relation of two points' are of course 
the reciprocals of those with respect to the harmonic relation of two lines, and do 
not need to be separately stated. 

The preceding theorems are useful in (among other geometrical investigations) the 
porism of the in-and-drcumscribed polygon. 

2 Stone Buildings, July 9, 1863. 



C. n. 



13 



98 



[119 



119. 



ON A THEOREM FOR THE DEVELOPMENT OF A FACTORIAL. 



[From the Philosophical Magnziney vol. vi. (1853), pp. 182—185.] 

The theorem to which I refer is remarkable for the extreme simplicity of its 

demonstration. Let it be required to expand the factorial x—a x — h x-^c ,.. in the 
form 



x — a x — x — y...-^- Bx" a a?--^... + Oa? — a ... + D ... &c. 
We have first 



a: — a = ar — a + a — a; 

multiply the two sides of this by a: — 6 ; but in multiplying by this factor the term 
.'/; — a, write the factor in the form a? — ^ + iS — 6 ; and in multiplying the term a — a, 
write the factor in the form x — a + a — b; the result is obviously 



x — ax— h^x — ax — fi +(a — a + /8 — 6)a? — a +a— aa— 6; 

multiply this by a; — c, this factor being in multiplying the quantity on the right-hand 

side written successively under the forms a: — 7 + 7 — c, a?— /8-I-/8 — c, a — a-^a — c] the 
result is 



X — a x^b X — c= x — ax — fix — y 



+ (a — a-hyS — 6 + 7 — c)a: — aa: — /8 
+ (a — a a— 6 + a-a /S — c-^-fi — b jS — c) x - a 



119] ON A THEOREM FOR THE DEVELOPMENT OF A FACTORIAL. 99 

which may be thus written, 
(x — a) (a? — 6) (ar — c) = 

Consider, for instance, 

La, 6, cJ, 

then, paying attention in the first instance to the Greek letters only, it is clear that 
the terms on the second side contain the combinations two and two, with repetitions, 
of the Greek letters a, ^, and these letters appear in each tenn in the alphabetical 
order. Each such combination may therefore be considered as derived from the primitive 
combination a, a by a change of one or both of the as into ^; and if we take 

(instead of the mere combination a, a) the complete first term a — a a — 6, and 
simultaneously with the change of the a of either of the factors into ^ make a similar 
change in the Latin letter of the factor, we derive from the first term the other terms 
of the expression on the right-hand side of the expression. It is proper also to 
remark, that, pa3dng attention to the Latin letters only, the different terms contain 
all the combinations two and two, without repetitions, of the letters a, 6, c. The same 
reasoning will show that 



x — ax — bx — cx — d== x-^a x — x — y x — 8 



La, 6, c, aJj 

la, b, c, dJ, 

[a. /8 -| 

+ x — a 

La, 6, c, dJ, 

La, b , c, dJt 



where, for instance, 



La, b, c, dit +(a-a)(a-b)(^-d) 



+ (a-a)(/8-c)(/9-d) 
+ (/9-6)(/3-c)(/9-d),&e. 



13—2 



100 ON A THEOREM FOR THE DEVELOPMENT OF A FACTORIAL. [119 

It is of course easy, by the use of subscript letters and signs of summation, to 
present the preceding theorem under a more condensed form ; thus writing 

LC&i U] ... Of , . . (lf.^gj f ^1 ^ ) 

where k,, Av-i, ...&o form a decreasing series (equality of successive terms not excluded) 

of numbers out of the system r, r — 1, ...3, 2, 1; the theorem may be written in the 
form 



X 






but I think that a more definite idea of the theorem is obtained through the notation 
first made use o£ It is clear that the above theorem includes the binomial theorem 
for positive integers, the corresponding theorem for an ordinary factorial, and a variety 
of other theorems relating to combinationa 

Thus, for instance, if Gg(ai,...ap) denote the combinations of a,, ... Op, q and q 
together without repetitions, and ^g (Oi, ... Op) denote the combinations of a,, ... Op, 
q and q together with repetitions, then making all the a's vanish, 



and therefore 



a?- tti ... aj-Op = ^^o(-)' C'gCoi* ••• ap)«*^r 
(X - a)P = SfoH' C, (a, a . . . plures) a^ = ^ S H' H a^ ^^. 



the ordinary binomial theorem for a positive and integral index p. 

So making all the a's vanish, 

a?** = /S^^o-ff9(«i ••• «p-«+i) ^^ *i ^"" ^ •••*■" "p-«- 

If m be any integer less than p^ the coefl&cient of x^ on the right-hand side 
must vanish, that is, we must have identically 

So also 

ffti . . . ci^«^+i I 
Cp-^((h, a,, ... Op) = Sqo (-)«Ci_^M»(«i» «» ••• «P-«) [^ ^J^- 

Suppose 

01 = 0, a, = l ...ap=2) — 1; ax=A, o, = i:— 1,... ap«fc-2)+ 1, 

then 

[:':..^J.-[":!.:.!:;:a-t^t*''^-'"^^-'''"^ 

and hence 

the binomial theorem for factorials. 



119] ON A THEOREM FOR THE DEVELOPMENT OF A FACTORIAL. 101 

A preceding formula gives at once the theorem 

It may be as well to remark^ with reference to a demonstration frequently given 
of the binomial theorem, that in whatever way the binomial theorem is demonstrated 
for integer positive indices, it follows from what has preceded that it is quite as easy 
to demonstrate the corresponding theorem for the factorial [m]^. But the theorem 
being true for the factorial [m]^, it is at once seen that the product of the series 
for (l+xy* and (1 +«?)** is identical with the series for (l+a:)*^+^ and thus it becomes 
unnecessary to employ for the purpose of proving this identity the so-called principle 
of the permanence of equivalent forms ; a principle which however, in the case in 
question, may legitimately be employed. 



102 



[120 



120. 



NOTE ON A GENERALIZATION OF A BINOMIAL THEOREM. 



[From the Philosophical Magazine, vol. vi. (1853), p. 185.] 

The formula {CrelUy t. i. [1826] p. 367) for the development of the binomial {x + a)*', 
but which is there presented in a form which does not put in evidence the law of 
the coefficients, is substantially equivalent to the theorem given by me as one of the 
Senate House Problems in the year 1851, and which is as follows : — 

"If {a 4-^8 + 7...)'' denote the expansion of (a 4-^ + 7... )p, retaining those terms 
Na^ff^rfh^ ,,. only in which 6 + c + d... is not greater than /) — 1, c-frf-f.. is not greater 
than p — 2, &c., then 

ar** = 1 (a: + o)** 

- "^"7'2W^^ {«-h/3-h7N^-i-a4-/3-h7 + 8)-. 
+ &c.' 



>i 



The theorem is, I think, one of some interest. 



121] 



103 



121. 



NOTE ON A QUESTION IN THE THEORY OF PROBABILITIES. 



[From the Philosophical Magazine^ vol. vi. (1853), p. 259.] 

The following question was suggested to me, either by some of Prof Boole's 
memoirs on the subject of probabilities, or in conversation with him, I forget which ; 
it seems to me a good instance of the class of questions to which it belongs. 

Given the probability a that a cause A will act, and the probability p that A 

acting the effect will happen; also the probability /8 that a cause B will su^t, and the 

probability q that B acting the effect will happen; required the total probability of 
the effect. 

As an instance of the precise case contemplated, take the following: say a day is 
called windy if there is at least w of wind, and a day is called rainy if there is at 
least r of rain, and a day is called stormy if there is at least W of wind, or if 
there is at least R of rain. The day may therefore be stormy because of there being 
at least W of wind, or because of there being at least R of rain, or on both 8w;counts ; 
but if there is less than W of wind and less than R of rain, the day will not be 
stormy. Then a is the probability that a day chosen at random will be windy, p the 
probability that a windy day chosen at random will be stormy, /8 the probability that 
a day chosen at random will be rainy, q the probability that a rainy day chosen at 
random will be stormy. The quantities X, fi introduced in the solution of the question 
mean in this psurticular instance, \ the probability that a windy day chosen at random 
will be stormy by reason of the quantity of wind, or in other words, that there will 
be at least W of wind; fi the probability that a rainy day chosen at random will 
be stormy by reason of the quantity of rain, or in other words, that there will be at 
least R of rain. 




104 NOTE ON A QUESTION IN THE THEORY OF PROBABILITIES. [l21 

The sense of the terms being clearly understood, the problem presents of course 
no difficulty. Let X be the probability that the cause A acting will act efficaciously ; 
fi the probability that the cause B acting will act efficaciously; then 

P = \ + (1-X)aa^. 
g = /[A + (l-/Lt)aX, 

which determine \, fi] and the total probability p of the effect is given by 

p = Xa + ^ - X/itt/S ; 
suppose, for instance, a=l, then 

;? = X + (l-X)/i^, g = /Lt+ X-X/Li, p = X + /Lt^-Xft/8, 

that is, p = p, for p is in this case the probability that (acting a cause which is 
certain to act) the effect will happen, or what is the same thing, p is the probability 
that the effect will happen. 

Machynlleth, August 16, 1853. 



122] 



105 



122. 



ON THE HOMOGRAPHIC TRANSFORMATION OF A SURFACE 

OF THE SECOND ORDER INTO ITSELF. 

[From the Philosophical Magazine, vol. vi. (1853), pp. 326—333.] 

The following theorems in plane geometry, relating to polygons of any number 
(odd or even) of sides, are well known. 

" If there be a polygon of (m + 1) sides inscribed in a conic, and m of the 
sides pass through given points, the (m + l)th side will envelope a conic having double 
coutact with the given conic." And " If there be a polygon of (m + 1) sides inscribed 
^ a conic, and m of the sides touch conies having double contact with the given 
^nic, the {m + l)th side will envelope a conic having double contact with the given 
conic.*' The second theorem of course includes the first, but I state the two separately 
for the sake of comparison vdth what follows. 

As regards the corresponding theory in geometry of three dimensions, Sir W. Hamilton 
given a theorem relating to polygons of an odd number of sides, which may be 
thus stated: "If there be a polygon of (2m + 1) sides inscribed in a surface of the 
^^ud order, and 2m of the sides pass through given points, the (2m + l)th side will 
constantly touch two sur&ces of the second order, each of them intersecting the given 
8urfece of the second order in the same four lines ^" 

^ See Phil, Mag. vol xxzt. [1S49] p. 200. The form in which the theorem is exhibited bj Sir W. Hamilton 
u somewhat different ; the surface containing the angles is considered as being an ellipsoid, and the two sorfaoes 
^'^'i^ hj the last or (2]ii + l)th side of the polygon are spoken of as being an ellipsoid, and a hjperboloid of 
two sheets, having respeotivelj doable contact with the given ellipsoid : the contact is, in fact, a qoadraple con- 
^ ^ the same four points ; real as regards two of them in the case of the ellipsoid, and as regards the other 
^ ift the case of the hyperboloid of two sheets ; and a qoadraple contact is the coincidence of foar generating 
^^>Mi belonging two and two to the two series of generating lines, these generating lines being of coarse (in the 
^Me Qoniidered by Sir W. Hamilton) all of them imaginary. 

c. n. 14 



106 ON THE HOMOQRAPHIC TBANSFOBMATION OF [122 

The entire theory depends upon Tvhat may be termed the transformation of a 
sur£BM^ of the second order into itself, or analytically, upon the transformation of a 
quadratic form of four indeterminates into itself I use for shortness the term trans^- 
formation simply; but this is to be understood as meaning a homographic transformation, 
or in analytic language, a transformation by means of linear substitutions. It will 
be convenient to remark at the outset, that if two points of a surface of the second 
order have the relation contemplated in the data of Sir W. Hamilton's theorem (viz. 
if the line joining the two points pass through a fixed point), the transformation is, 
using the language of the Recherchss Arithm^tiqtiea, an improper one, but that the 
relation cont-emplated in the conclusion of the theorem (viz. that of two points of a 
surface of the second order, connected by a line touching two surfaces of the second 
order each of them intersecting the given sur&ce of the second order in the same 
four lines) depends upon a proper transformation; and that the circumstance that an 
even number of improper transformations is required in order to make a proper trans- 
formation (that this circumstance, I say), is the reason why the theorem applies to 
polygons in which an even number of sides pass through fixed points, that is, to 
polygons of an odd number of sides. 

Consider, in the first place, two points of a sur£ace of the second order such that 
the line joining them passes through a given point. Let a, y, z, w he current 
coordinates S and let the equation of the surface be 

(a,. ..)(«, y, z, wy = 0, 

and take for the coordinates of the two points on the surface Xi, yi, Zi, v\ and 
^i> Vti ^s> ^1} ai^d for the coordinates of the fixed point a, /3, 7, S. Write for shortness 

(a, ...)(«, /S, 7, S)«=p, 
(a, ...)(«, ^, 7, S)(a?i, yi, z^, Wi) = qu 
then the coordinates ^, y,, z^, w^ are determined by the very simple formulsB 

2ce 

27 

P 

2S 

^ Strictly speaking, it is the ratios of these qoantities, e.g. « : tr, y : 19, z : w, whioh are the ooordinates, and 
consequently, even when the point is given, the values «, y, «, to are essentially indeterminate to a factor prh. 
So that in assu mi ng that a point is given, we should write xiyiz: w=a : P'.yiH; and that when a point is 
obtained as the result of an analytical process, the conclusion is necessarily of the form just mentioned : but 
when this is once understood, the language of the text may be properly employed. It may be proper to explain 
here a notation made use of in the text: taking for greater simplicity the case of forms of two variables, 
(I, m) {x^ y) means te+my ; (a, 6, e) («, y)« means a«*+26xy+cy«; {a,b,e) ft, iy) (ar, y) means a^x + h{fy + rtx) + e7fy. 
The system of coefficients may frequently be indicated by a single coefficient only : thus in the text (a, ...) («, y, f , ir)^ 
stands for the most general quadratic ftmotion of four variables. 



122] 



A SUBFACE OF THE SEOOND ORDER INTO ITSELF. 



107 



In &ct, these values satisfy identically the equations 

«i, y%, ^, w% 



0, 



«i, Vu Sk, Wi 

a, 0, y, S 

that is, the point (xt, y%, z^, w^ will be a point in the line joining {Xi, j/i, z^ Wi) 
and (a, fi, y, S). Moreover, 

(a, ...)(«!, yi» -^1, w«)*= (a» ...)(«^i. Vu ^» ^y 

--^ (a, ...)(«, A 7, S)(«i, yi, Zu Wi) 

(a, ...) («i, yi, ir^, t^i)« - -^' ?i + % i>, 
that is, 

80 that d?|, yi, Zi, Wi being a point on the surface, a?„ y,, z^, w^ will be so too. The 
equation just found may be considered as expressing that the linear equations are a 
transformation of the quadratic form (a, ...)(a;, y, z, vif into itself If in the system 
of linear equations the coefficients on the right-hand side were arranged square-wise, 
and the determinant formed by these quantities calculated, it would be found that 
the value of this determinant is —1. The transformation is on this account said to 
be improper. If in a system of linear equations for the transformation of the form 
mto itself the determinant (which is necessarily -h 1 or else — 1) be +1, the trans- 
formation is in this case said to be proper. 

We have next to investigate the theory of the proper transformations of a quadratic 
form of four indeterminates into itself This might be done for the absolutely general 
form by means of the theory recently established by M. Hermite, but it will be 
sufficient for the present purpose to consider the system of equations for the trans- 
f(Mination of the form aj*-hy' + -2^ + ti;* into itself given by me some years since. (Crdle, 
voL xxxn. [1846] p. 119, [52] Q). 

I proceed to establish (by M. Hermite's method) the formulae for the particular 
case in question. The thing required is to find ^, y„ z^, w^ linear functions of 
"hf Vu ^t ^u such that 

^' + y«* + z^^ -I- 1^,« = a?i* + yi« + -^1* + Wi*. 
Write 

flr,+«i=2f, yi + y, = 2i7, Zi + z^:=2Z ii;i + w, = 2o); 

^ It is a jjingnlar instanoe of the way in which different theories connect themselves together, that the 
toannls in qoeetion were gooeralizations of Baler's formolaB for the rotation of a solid body, and also are 
loaaavlm which reappear in the theory of qaatemions ; the general formolas cannot be established by any obyions 
ggpsraliaation of the theory of qaatemions. 

14—2 



108 ON THE HOMOGRAPHIC TRAJ^SFOEMATION OF [122 

then putting a:, = 2f — a?i, &c., the proposed equation will be satisfied if only 

f" + if + (7+ «*= f^ + Wi + f^i + a)Wi, 
which will obviously be the case if 

Zi = /Ltf — Xiy + ? + ccD , 
Wi = — af — 617 — c(^ + a , 
where X, /li^ 1^, a, b, c are arbitrary. 

Write for shortness 

a\ + bfjL + cv^il>, l + X« + /Lt« + i/» + a« + 6' + (^ + 0« = A;, 
then we have 
A:f = (l +X*+ 6"+ (^)a^ + (XM-y -a5-c0)yi + (i'X. + /i-ca + 6^)^i + (6i;-c/i-a-\^)wi, 

A:(;' = (i'X-/Lt-ca-6^)a?i + (AM/ + X — 6c +a^)yi+(l +i^ + a*+fe» )-?!+ (aft—ftX.- 0-1/^)1^1, 
A;u;= (61/ - c/i+ a +\^) a?i + (cX. -ay+ 6 +/A^)yi+ (a^ -6*'+ c + 1/^) ^1 + (1 + X* +/Lt» + i^) i^/i ; 
and fix>m these we obtain 

Auji = (1 + X» + 6« + (^ - /Lt*- 1/'- a' - ^)a?i + 2 (X/Li-i/- oft- c<^)yi + 2 (vX. + M-ca + 6^) ^1 

+ 2 (6v — c/Li — a— X^) Wi, 

A:ya = 2(X/Lt + y-a6 + c0)a?i + (l + M* + (^ + a«-i^-X«-fe«-<^)yi4-2(/[Ay-X-6c-a^)^^ 

+ 2 (cX — ai/ — 6 — /Lt^) w/j , 

A:-?a = 2(i/X-/Lt-ca-6^)a?i + 2(/Ay + X-6c + a^)yi + (l + i/» + a» + 6*-X«-/Lt*-c>-«^)^^ 

+ 2 (a/Li — 6X — c — y^) Wi , 

Arwj = 2 (61/ — c/Li + a + X^) a?i + 2 (cX - ai/ + 6 + ft^) yi + 2 (a/Li — 61/ + c + p4^)^i 

+ (1 + X» +/iA« + i;»-a« -6» - c"- 0«) Wi, 

values which satisfy identically x^ -V yf •\' zf •\- w^ ^ x^ •\- y^ '\- z^ •\- w^. 

Dividing the linear equations by &, and forming with the coefficients on the right- 
hand side of the equation so obtained a determinant, the value of this determinant is 
4- 1 ; the transformation is consequently a proper one. And conversely, what is very 
important, every proper transformation may be exhibited under the preceding form*. 

I The nature of the reasoning by which this is to be established may be seen by considering the analogous 
relation for two variables. Suppose that x^, y^ are linear functions of x and y such that x^-{-y^=a^+y'*\ then 
if 2f=« + ari, 2iy=y+yi, ^, 17 will be linear functions of ar, y such that ?+i;*=$«+i;y, or $(f-«) + i7(i7-y) = 0; 
^-x must be divisible either by ri or else by 17-y. On the former supposition, calling the quotient y, we have 
:c=(-Fi}, and thence ^=^(+17, leading to a transformation such as is considered in the text, and which is a proper 
transformation; the latter supposition leads to an improper transformation. The given transformation, assumed 
to be proper, exists and -cannot be obtained from the second supposition; it must therefore be obtainable from 
the first supposition, i.e. it is a transformation which may be exhibited under a form such as is considered in 
the text. 



122] A SURFACE OF THE SECOND ORDER INTO ITSBLF. 109 

Next considering the equations connecting x, y, z, w with {, 97, (^, od, we see that 

+ ( Mf-A'^+ ?+Cft))» 
+ (-af -617- cf+ ft))*. 

We are thus led to the discussion (in connexion with the question of the trans- 
formation into itself of the form a" + y* + ^ + ti;*) of the new form 

+ (--vx+ y + 7iz + bvif 
+ ( /jLx — \y+ z + cwy 
+ (— oo? — 6y — c-^4- w)"; 

or, as it may also be written, 

(af + y* + j^-{'tul') + (vy-fiz-{'awy-{-(\z — vx + bwy + (jix-\y-\'C^ 

Represent for a moment the forms in question by U, F, and consider the surfaces 
[7=0, F=0. If we form from this the sur&ce V+qU=0, and consider the dis- 
cnminant of the function on the left-hand side, then putting for shortness 

/c = V + /Lt« + i/« + a« + 6»+c>, 
this discriminant is 

which shows that the sur&ces intersect in four lines. Suppose the discriminant vanishes; 
we have for the determination of 9 a quadratic equation, which may be written 

5« + (2 + /e)5 + ir=0; 

let the roots of this equation be q,, q,/, then each of the functions q,U-\-V, q,,U-^ V 
will break up into linear factors, and we may write 

q,U+V^RA> 

q,,U+r^RAr 

(U and V are of course linear functions of R^, and R„8,^) forms which put in 
evidence the fact of the two sur&ces intersecting in four lines. 

The equations 

a^ + a?, = 2f, yi+yt«2i;, Zi+Zi^2^, Wi + t£;, = 2fii, 



no ON THE HOMOGRAPHIC TRANSFORMATION OF [122 

show that the point (f, 17, f, «) lies in the line joining the points (a^i, yi, z^ Wi) and 
(^11 3/21 ^21 wjj); aiid to show that this line touches the surface F=0, it is only 
necessary to form the equation of the tangent plane at the point (f, 1;, (Ti <») o{ the 
8ur£BU3e in question ; this is 

(x + vy --fjLZ + at(;) (f + V17 — ft(r + ck») + ... = ; 
or what is the same thing, 

{x •\- vy "' fiz •¥ aw) a?i + . . . = 0, 

which is satisfied by writing (a?,, yi, i^i, Wj) for (x, y, -e, w), that is, the tangent plane of 
the surface contains the point (xi, y^ Zi, w,). We see, therefore, that the line through 
(^, yii ^1. «^i) aiid (^> yjj -8^2 » ^2) touches the surface F=0 at the point (f, 17, f, a>). 

Write now 

^=~a'' ^=~r"' ^=":r» ^=^"a"» f^'^'iT' ^="^5 
<p 9 9 9 9 9 

if we derive from the coordinates a?!, yi, ^Ti, Wi, by means of these coefficients 
a\ b\ c\ \'y f/, v\ new coordinates in the same way as «i, y„ z^ w^ were derived by 
means of the coefficients a, 6, c, X, /li, r, the coordinates so obtained are — a?i, — yi,~-z^i, — Wi, 
i.e. we obtain the very same point (x^, y,, ^si ^i) by means of the coefficients (a, 6, c, X, /a, i;), 
and by means of the coefficients (a\ h\ c\ \\ /li', v^). Call f, rf, f, oi' what f, 17, f, c» 
become when the second system of coefficients is substituted for the first; the point 
^j V> (r'» ^' will be a point on the sur&oe F' = 0, where 

F'=<^>(a;» + y« + ^ + ti;») 

+ (— cy + 62: — Xw)* + (— 0^ -i: ca? — /it^;)* + (— 6a: + ay — j/u;)" + (— Xr — /Lty — i/^)* ; 
and since 

F+ F' = /e(a» + y« + j^» + t£;»), 

and F=0 intersects the surface iB* + y* + ^ + tt;* = in four lines, the surface F' = 
will also intersect this surface in the same four lines. And it is, moreover, clear that 
the line joining the points {x^, yi, Zi, Wi) and (x^, y,, ^„ n;,) touches the surfieu^ F' = 
in the point (f', 17', f, co'). We thus arrive at the theorem, that when two points 
of a surfisu^ of the second order are so connected that the coordinates of the one 
point are linear functions of the coordinates of the other point, and the transformation 
is a proper one, the line joining the two points touches two sur&ces of the second 
order, each of them intersecting the given surface of the second order in the same 
four lines. Any two points so connected may be said to be corresponding points, or 
simply a pair. Suppose the four lines and also a single pair is given, it is not for 
the determination of the other pairs necessary to resort to the two auxiliary surfaces 
of the second order ; it is only necessary to consider each point of the surface as 
determined by the two generating lines which pass through it; then considering first 



122] A SURFACE OF THE SECOND ORDER INTO ITSELF. Ill 

one point of the given pair, and the point the corresponding point to which has to 
be determined, take through each of these points a generating line, and take also 
two generating lines out of the given system of four lines, the four generating lines 
in question being all of them of the same set, these four generating lines inter- 
secting either of the other two generating lines of the given sjrstem of four lines in 
four pointa Imagine the same thing done with the other point of the given pair 
and the required point, we should have another system of four points (two of them 
of course identical with two of the points of the first-mentioned system of four points); 
these two systems must have their anharmonic ratios the same, a condition which 
enables the determination of the generating line in question through the required 
point: the other generating line through the required point is of course determined 
in the same manner, and thus the required point (i.e. the point corresponding to any 
point of the surface taken at pleasure) is determined by means of the two generating 
lines through such required point. 

It is of course to be understood that the points of each pair belong to two 
distinct systems, and that the point belonging to the one system is not to be con- 
founded or interchanged with the point belonging to the other system. Consider, now, 
a point of the surface, and the line joining such point with its corresponding point, 
but let the corresponding point itself be altogether dropped out of view. There are 
two directions in which we may pass along the surface to a consecutive point, in 
such mamier that the line belonging to the point in question may be intersected by 
the line belonging to the consecutive point. We have thus upon the surface two 
series of curves, such that a curve of each series passes through a point chosen at 
pleasure on the surface. The lines belonging to the curves of the one series generate 
a series of developables, the edges of regression of which lie on one of the surfaces 
intersecting the surfeice of the second order in the four given lines; the lines belonging 
to the curves of the other series generate a series of developables, the edges of 
regression of which lie on the other of the surfaces intersecting the surface in the 
four given lines; the general nature of the system may be understood by considering 
the system of normals of a surface of the second order. Consider, now, the surface 
of the second order as given, and also the two surfaces of the second order inter- 
secting it in the same four lines ; from any point of the sur&ce we may draw to 
the auxiliary surfiEtces four dififerent tangents ; but selecting any one of these, and 
considering the other point in which it intersects the surface as the point corre- 
sponding to the first*mentioned point, we may, as above, construct the entire system 
of corresponding points, and then the line joining any two corresponding points will 
be a tangent to the two auxiliary surfSaces; the system of tangents so obtained may 
be called a system of congruent tangents. Now if we take upon the surface three 
points such that the first and second are corresponding points, and that the second 
and third are corresponding points, then it is obvious that the third and first are 
corresponding points ;— observe that the two auxiliary surfaces for expressing the corre- 
spondence between the first and second point, those for the second and third point, 
and those for the third and first point, meet the surface, the two auxiliary surfaces 
of each pair in the same four lines, but that these systems of four lines are different 



112 ON THB HOMOGRAPHIC TRANSFOBMATION OF A SURFACE &C. [122 

for the different pairs of auxiliary surfaces. The same thing of course applies to any 
numb^ of corresponding points. We have thus, finally, the theorem, if there be a 
polygon of (m + 1) sides inscribed in a surfiEtce of the second order, and the first side 
of the polygon constantly touches two surfaces of the second order, e8w;h of them 
intersecting the surfisu^ of the second order in the same four lines (and the side 
belong always to the same system of congruent tangents), and if the same property 
exists with respect to the second, third, &c.... and wth side of the polygon, then will 
the same property exist with respect to the (m + l)th side of the polygon. 

We may add, that, instead of satisfying the conditions of the theorem, any two 
consecutive sides of the polygon, or the sides forming any number of pairs of con- 
secutive sides, may pass each through a fixed point. This is of course only a 
particular case of the improper transformation of a surfietce of a second order into 
itself, a question which is not discussed in the present paper. 



1231 



113 



123. 



ON THE GEOMETRICAL REPRESENTATION OF AN ABELIAN 

INTEGRAL. 



[From the Philosophical Magazine, vol. vi. (1853), pp. 414 — 418.] 

The equation of a surface passing through the curve of intersection of the surfaces 

a^+ y*+ -^'+ u;* = 0, 
flw^ + 6y* + cz^ H- dv)* = 0, 
is of the form 

»(a^ + y' + -2' + t£;') + aa^ + 6y' + c^ + dti;* = 0, 

where tt is an arbitrary parameter. Suppose that the sur&ce touches a given plane, 
we have for the determination of 8 a cubic equation the roots of which may be 
considered as parameters defining the plane in question. Let one of the values of 8 
be considered equal to a given quantity k, the plane touches the surface 

aud the other two values of 8 may be considered as parameters defining the particular 
tangent plane, or what is the same thing, determining its point of contact with the 
Burface. 

Or more clearly, thus: — in order to determine the position of a point on the 
8ur£BM^e 

A; (a;* + y2 + ^* + w") + aa?» + 6y« + C2:« + dt(;» = ; 

the tangent plane at the point in question is touched by two other surfaces 

/> (^ + y ' + ^ + 1^;") + cue" + 6y * + c^* + dw" = 0, 

9(«' + y* + '8^* + w*) + cuc» + 6y' + C2:* + dt£;» = 0; 

c. n. 15 



114 ON THE GEOMETRICAL REPRESENTATION OF AN ABELIAN INTEGRAL. [l23 

and, this being so, p and q are the parameters by which the point in question is 
determined. We may for shortness speak of the surface 

A: (iB» + y» + ^ + w") + oa^ + 6^ + c-e' + dw" = 
as the surfiiK^ (jk). It is clear that we shall then have to speak of 



ai^ + y^ + z^ + v)^ = 



as the surface (oo). 



I consider now a chord of the surface (oo) touching the two surfaces (k) and 
(k')] and I take 0, <l> aa the parameters of the one extremity of this chord; (p, q) 
as the parameters of the point of contact with the surface (A;); p', q' as the parameters 
of the point of contact with the surface (kf); and O', ^' as the parameters of the 
other extremity of the chord; the points in question may therefore be distinguished 
as the points (oo ; ^, 0), (i; p, g), (k' \ p\ g'), and (oo; ^, ^'). The coordinates of the 
point (oo ; 0, <f>) are given by 

X : y : z : w= V(a + ^) (a + 0) -:- V(a — 6) (a — c) (a - d) 

V(6+^)(6 + </>) -r V(6-c)(6-d)(6-a) 

V(c + ^) (c + 4>) -r V(c - d) (c - a) (c - 6) 

^/(dTW(d + 'f) H- V(d-a)(d-6)(d-c) ; 
those of the point (k; p, q) by 

X ', y : z : w^ V(a +p) (a + j) -^ V(a — 6) (a — c) (a — d) Va + i 

V(6+p)(6 + 5) -^ V'(6-c)(6-d)(6-a) ViTib 

V(c + p) (c + g) -r V(c - d) (c - a) (c - 6) Vc+i 
V(d+/>)(d + g)-f.V(d-a)(d-6)(d-c) Vd+l; 
and similarly for the other two points. 

Consider, in the first place, the chord in question as a tangent to the two 
surfaces (£) and {kf). It is clear that the tangent plane to the surface {k) at the 
point {k\ p, q) must contain the point (Ar'; p\ <f)^ and vice versd. Take for a moment 
f, i;, (r> ® ®* *^® coordinates of the point (A; p, g), the equation of the tangent 
plane to (£) at this point is 

2(a + A:)fd: = 0; 

or substituting for f,... their values 

S (a;V(a+p)(a + g) Va -{Ic -r- V(a - 6) (a - c) (a - d) ) = ; 



123] ON THE GEOMETRICAL REPRESENTATION OF AN ABELTAN INTEGRAL. 115 

or taking for a?,... the coordinates of the point (kf^ p\ ^\ we have for the conditions 
that this point may lie in the taDgent plane in question, 

or under a somewhat more convenient form we have 

2((6-c)(c-d)(d-6)V(a+p)(a + ?)V(a + p')(a4-gO^^) = 0, 

for the condition in order that the point (Je\ p\ q') may lie in the tangent plane at 
(^> P* ?) ^ *^® surface {k). Similarly, we have 

2f(6-c)(c-cO(d-6)V(a+i>)(a + g)V(a+/)0(a + 9')^5^')=O, 
\ ya-\- kJ 

for the condition in order that the point (A?, p, q) may lie in the tangent plane at 
ifff\ p\ 5O ^ *^® surfiwje Qf). The former of these two equations is equivalent to 
the sjrstem of equations 

V(a +/>) (a + q) (a +;>0 (« + 9') V l^A/ = ^ + A^ + ^'> 

and the latter to the system of equations 

V(o +p) (o + q) (a + p') (a + q) y |^ = V + /t'o + v'a*; 

where in each system a is to be successively replaced by 6, c, d, and where X, /ia, 1/ 
and V, ;i', 1/ are indeterminate. Now dividing each equation of the one system by 
the corresponding equation in the other system, we see that the equation 

x-\-k _ \ -{- fix + va^ 

^ satisfied by the values a, h, c, d oi x\ and, therefore, since the equation in a? is 
^^y of the third order, that the equation in question must be identicaUy true. We 
naay therefore write 

\ + fix-\-va^=^(px-^<T)(x + k\ \' + fix+v*a^ = (px + a)(x-\'^), 
*^^ the two systems of equations become therefore equivalent to the single system, 

V(a +p) (a + q) (a +/) (a -f q') = (pa + a) V(a + *)(« + *'), 
V(6+;?)(6 + 9)(6+;?')(6 + ?') = (p6 + <^) V(ft + A) (6 + A:'), 
V(c +p) (c + q) (c + Jt>0 (c + 9') = (pc + cr) V(c + A) (c + A:'), 

V(d +i>) (d + ?) (d +p') {d -h gO = (pd + cr) V(d + A:) (d + jfc'), 

15—2 



116 ON THE GEOMETRICAL REPRESENTATION OF AN ABELIAN INTEGRAL. [123 

a set of equations which may be represented by the single equation 
where x is arbitrary; or what is the same thing, writing —a: instead of a?, 

Hence, putting 

J v(a? + a)(a: + 6)(a: + c)(a? + cO(a?-A:)(a?-A?')* 

J V(a?4-a)(a? + 6)(a? + c)(a? + c0(a?-A:)(a?-ifc') 

we see that the algebraical equations between p, q\ p\ <]( are equivalent to the 
transcendental equations 

Up ±liq ± Up' ± Ilgr' = const. 
TI,p ± U^q ± ny ±U,q' = const. 

The algebraical equations which connect 0, <f> with p, q; p\ ((, may be exhibited 
under several different forms; thus, for instance, considering the point (oo ; ^, </>) as 
a point in the line joining (A; p, q) and (A:'; p\ q% we must have 



V(a+p)(a + g) ^ VoTA, V(6 4-p)(6 + g') -^ \/6 + *,. . . 



= 0, 



i.e. the determinants formed by selecting any three of the four columns must vanish; 
the equations so obtained are equivalent (as they should be) to two independent 
equations. 

Or, again, by considering (oo ; d, ^) first as a point in the tangent plane at 
{k\ p, q) to the surface (k), and then as a point in the tangent plane at (Ar^; p\ q') 
to the surface (&'), we obtain 

2 ((6- c)(c-d)(d - 6) V(a + p)(a + g) V(aTl) V(a + ^)(a4-<^)) = 0, 

2((6-c)(c-(0(d-6)V(^TpOMY^ 

Or, again, we may consider the line joining (oo ; 6, <f>) and (k; p, q) or (A/; p\ q'), 
as touching the surfaces (k) and (A/); the formulae for this purpose are readily 
obtained by means of the lemma, — 



123] ON THE GEOMETRICAL REPRESENTATION OF AN ABELIAN INTEGRAL. 117 

" The condition in order that the line joining the points (f , i;, (^, ©) and (f', n\ ^, (o') 
inay touch the surface 

is 

2ab(fi7'-r^)^ = 0. 
the summation extending to the binary combinations of a, b, c, dr 

But none of all these formulae appear readily to conduct to the transcendental 
equations connecting 0, with p, q; p\ q\ Reasoning from analogy, it would seem 
that there exist transcendental equations 

±U0 ±U<f} tllp ± Up' =con8t. 
± n^0 ± n,<f> ± U^p ± U^p' = const., 

or the similar equations containing q, ((, instead of j}, p\ into which these are changed 
by means of the transcendental equations between j), q, p\ ((. If in these equations 
we write ff^ (f/ instead of 0, ^, it would appear that the functions Up, lip', II^p, Tl^p' 
may be eliminated, and that we should obtain equations such as 

±T10 ±U(I> ±U0' ± n<f)' = const. 
± U,0 ± U^<f> ± n,^ ± n,</)' = const. 

to express the relations that must exist between the parameters 0, <(> and ^, <!>' of 
the extremities of a chord of the surface 

ic» + y» + ^ + t(;* = 0, 

b order that this chord may touch the two surfaces 

* (^ H- y* + ^ + 1^;") + our* H- 6y* + c^ + dt£;» = 0, 
Ar'(aj» + y» + ^ + «(;») -f aa;* H- 6y* H- c^ + dw" = 0. 

The quantities k, V, it will be noticed, enter into the radical of the integrals 
JIo?, II/p. This is a very striking difiference between the present theory and the 
analogous theory relating to conies, and leads, I think, to the inference that the theory 
of the polygon inscribed in a conic, amd the sides of which tovdt conies intersecting 
the conic in the same four points, cannot be extended to surfaces in such manner as 
one might be led to suppose from the extension to surfaces of the much simpler 
theory of the polygon inscribed in a conic, and the sides of which totich conies having 
double conta^ct with the conic, (See my paper "On the Homographic Transformation 
of a 8ur£Eu;e of the second order into itself," [122]). 

The preceding investigations are obviously very incomplete; but the connexion 
which they point out between the geometrical question and the Abelian integral 
involving the root of a function of the sixth order, may I think be of service in 
the theory of these integrals. 



118 



[124 



124 



ON A PEOPEETY OF THE CAUSTIC BY EEFEACTION OF THE 

CIECLE. 



[From the Philosophical MagcLzine, vol vi. (1853), pp. 427 — 431.] 

M. St Laurent has shown (Oergonne, vol. xviu. [1827] p. 1), that in certain cases 
the caustic by refiraction of a circle is identical with the caustic of reflexion of a circle 
(the reflecting circle and radiant point being, of course, properly chosen), and a very 
elegant demonstration of M. St Laurent's theorems is given by M. Gergonne in the 
same volume, p. 48. A similar method may be employed to demonstrate the more 
general theorem, that the same caustic by refraction of a circle may be considered as 
arising from six different systems of a radiant point, circle, and index of refr'action. 
The demonstration is obtained by means of the secondary caustic, which is (as is well 
known) an oval of Descartes. Such oval has three foci, any one of which may be 
taken for the radiant point: whichever be selected, there can always be found two 
corresponding circles and indices of refraction. The demonstration is as follows : — 

Let c be the radius of the refiucting circle, /a the index of refraction; and taking 
the centre of the circle as origin, let f, 17 be the coordinates of the radiant point, 
the secondary caustic is the envelope of the circle 



where a, /3 are parameters which vary subject to the condition 

the equation of the variable circle may be written 

{^2(a^ + y« + c«)-(f> + i7« + c')}-2(/i«a:-f)a-2(/i»y-i7)/9 = 0, 



124] ON A PROPERTY OP THE CAUSTIC BY REFRACTION OP THE CIRCLE. 119 
which is of the form 

the envelope is therefore 

Hence substituting, we have for the equation of the envelope, Le. for the secondary 
caustic, 

(A^^a^ + y' + c») - (P + 17' + c»)}» = 4c» {(Ai'a; - f )« + (M»y - 17)»}, 
which may also be written 

and this may perhaps be considered as the standard form. 

To show that this equation belongs to a Descartes' oval, suppose for greater con- 
venience 17 = 0, and write 

/Lt*(a5» + y"- c*) - p + c* = 2c/i V(a?-f)» + y» ; 

1 . / IV 
multiplying this equation by 1 — 5, and adding to each side c^lfi ) 4.(3? — f)* + y«, 

we have 

(i-^.){M»(a^+y'-c)-?+c}+(«-f)'+y'+c(A*-iy 

= (a;-f)» + y» + 2c(/t*-i)V(x-f)» + y' + c»(M-y*; 
0' Kdacing 

Bgun, multiplying the same equation by — (l — 5), and adding to each side 

we have 

^' reducing, 

(«_^' + y. = ||V(a,_f). + y. + |(l-Dp. 



120 ON A PROPBRTY OP THE CAUSTIC BY REFRACTION OF THE CIRCLE, [124 

Hence, extracting the square roots of each side of the equations thus found, we 
have the equation of the secondary caustic in either of the forms 






to which are to be joined 



Vi'-^'- 


^^=f\/( 


.-'-M'j ^y 




c(.-y7(.-g.^.(- 


f.^y(. 


-^hy'' 


/fr fA.^\ 


.(|_^)V(«;-f). + y. 


Write successively, 








r=f . 


c' = c , 


/t*' = M , 


(1) 




A* 


/ c 


(«) 






^■'\- 


(/9) 


r=f . 




"'-? • 


(7) 




c' = Cf 


"-T' 


(«) 








(0 


or, what is the same thing. 








f=r . 


c = c', 


^=/*' . 


(1) 








(«) 




c=-, 


1 


(/3) 


f=r, 




r 


(7) 




CSC*, 




(8) 


V 1:' > 


(Z 

''=/• 


c' 


(«) 



r=f . 






H' 


- =? 


A*" ^ ' 


H- 






r=f . 



















124] ON A PROPERTY OF THE CAUSTIC BY REFRACTION OF THE CIRCLE. 121 
or, again, 

(1) 
(«) 

(7) 
(8) 

(*) 

then, whichever system of values of f, c', /Lt' be substituted for f, c, /a, we have in 
each case identically the same secondary caustic, the effect of the substitution being 
amply to interchange the different forms of the equation; and we have therefore 
identically the same caustic. By writing 

&c., 

^ j9, 7, 5, € will be functional symbols, such as are treated of in my paper " On the 
Theory of Groups as depending on the symbolic equation 6^ = 1," [126], and it is 
^ to verify the equations 

= ^8" = 87 = eS = 7€, 
/9s= a? = ey = 78 = Se, 
7 = Sa = 6/9 = /98 = oe, 
S = ea = 7/8 = 07 = /Se, 
e = 7a = 8/9=/97 = aS. 

Suppose, for example, f= — c, i.e. let the radiant point be in the circumference; 
™n in the fourth system ^ = — c, c' = — , (or, since d is the radius of a circle, this 

radius may be taken - ), /Lt' = — 1, or the new system is a reflecting sjrstem. This is 
^^^ of M. St Laurent's theorems, viz. 

C. IL 16 



122 ON A PROPBRTY 0*" THE CAUSTIC BY REFRACTION OF THE CIRCLE. [124 

Theorem. The caustic by refiractiou of a circle when the radiant point is on the 
circumference, is the caustic by reflexion for the same radiant point, and a concentric 
circle the radius of which is the radids of the first circle divided by the index of 
re&action. 

Again, if f = — c/li, the fifth system gives ^ = ^^ , d — Cy /a' =^ — 1, or the new system 

is in this case also a reflecting system. This is the other of M. St Laurent's 
theorems, viz. : — 

Theorem. The caustic by refraction of a circle when the distance of the radiant 
point bom the centre is equal to the radius of the circle multiplied by the index of 
refiraction, is the caustic by reflexion of the same circle for a radiant point which Ls 
the image of the first radiant point. 

Of course it is to be understood that the image of a point means a point whose 
distance firom the centre = square of radius -r- distance. 

2 Stom Buildings, Nov. 2, 1863. 



125] 



123 



125. 



ON THE THEORY OF GROUPS, AS DEPENDING ON THE 

SYMBOLIC EQUATION ^=1. 

[From the Philosophical Magazine, vol. vii, (1854), pp. 40 — 47.] 

Let ^ be a symbol of operation, which may, if we please, have for its operand, 
not a single quantity x, but a sjrstem (x, y, ...)> ^^ ^^^^ 

0(^, y, ...) = («^. y', ...), 

where a/, y', ... are any functions whatever of x, y, ..., it is not even necessary that 
x\ y\ ... should be the same in number with x, y, ..,. In particular a/, y', &c. may 
represent a permutation of x, y, &c., is in this case what is termed a substitution; 
and if^ instead of a set x, y, ..., the operand is a single quantity x, so that Ox^af =fx, 
fl is an ordinary functional symbol. It is not necessary (even if this could be done) 
to attach any meaning to a symbol such as tf ± <^, or to the symbol 0, nor con- 
sequently to an equation such as ^ = 0, or ^±^ = 0; but the symbol 1 will naturally 
denote an operation which (either generally or in regard to the particular operand) 
leaves the operand unaltered, and the equation = <l> will denote that the operation 
is (either generally or in regard to the particular operand) equivalent to kJ), and 
of course 0=1 will in like manner denote the equivalence of the operation to the 
operation 1. A sjnmbol 0<l> denotes the compound operation, the performance of which 
is equivalent to the performance, first of the operation (f>, and then of the operation 
0; 0^ is of course in general different from ^^. But the symbols 0, 0, ... are in 
general such that ^ . ^ = ^0 . %> &c., so that 0^, 0<lyx^t ^^- ^^^^ ^ definite signi- 
fication independent of the particular mode of compounding the symbols; this will 
be the case even if the functional operations involved in the sjmcibols 0, ^, &c. 
contain pajrameters such as the quaternion imaginaries i, j, k; but not if these 
iiinctional operations contain parameters such as the imaginaries which enter into the 
theory of octaves, &c., and for which, e.g. a.l3y is something different from a^.7, 

16—2 



124 



ON THE THEORY OF GROUPS, 



[125 



a supposition which is altogether excluded from the present paper. The order of the 
factors of a product ^^;^... must of course be attended to, since even in the case 
of a product of two factors the order is material; it is very convenient to speak of 
the symbols 6, ^ . . . as the first or furthest, second, third, &c., and last or nearest 
factor. What precedes may be almost entirely summed up in the remark, that the 
distributive law has no application to the symbols ^^ ... ; and that these symbols are 
not in general convertible, but are associative. It is easy to see that ^=1, and 
that the index law ^.^=6?^+'*, holds for all positive or negative integer values, 
not excluding 0. It should be noticed also, that if tf = 0, then, whatever the symbols 
o, /8 may be, a0ff = a<f>0, and conversely. 

A set of symbols, 

1, a, ^, ... 

all of them different, and such that the product of any two of them (no matter in 
what order), or the product of any one of them into itself, belongs to the set, is 
said to be a groups. It follows that if the entire group is multiplied by any one 
of the symbols, either as further or nearer factor, the effect is simply to reproduce 
the group; or what is the same thing, that if the symbols of the group are multi- 
plied together so as to form a table, thus: 

Farther factors 

1 a /3 .. 



to 

a 



13 



1 

a 


a 


)8« 


a* 


/9 

• 
• 


«/3 


/S* 





that as well each line as each column of the square will contain all the symbols 

1, a, ^, — It also follows that the product of any number of the S3mibols, with or 

Avithout repetitions, and in any order whatever, is a symbol of the group. Suppost^ 

that the group 

1, a, ^, ... 

contains n symbols, it may be shown that each of these sjrmbols satisfies the equation 

^ = 1; 

so that a group may be considered as representing a system of roots of this symbolic 
binomial equation. It is, moreover, easy to show that if any symbol o of the group 

1 The idea of a group as applied to permatations or substitutions is due to Oalois, and the introduction 
of it may be considered as marking an epoch in the progress of the theory of algebraical equations. 



125] 



AS DEPENDING ON THE SYMBOLIC EQUATION ^=1. 



125 



satisfies the equation ^ = 1, where r is less that n, then that r must be a sub- 
multiple of n; it follows that when n is a prime number, the group is of necessity 
of the form 

1, a, a\...a^-\ (a«=l); 

and the same may be (but is not necessarily) the case, when n is a composite 
number. But whether n be prime or composite, the group, assumed to be of the 
form in question, is in every . respect analogous to the system of the roots of the 
ordinary binomial equation a:* — 1 = ; thus, when n is prime, all the roots (except 
the root 1) are prime roots; but when n is composite, there are only as many prime 
roots as there are numbers less than n and prime to it, &c. 

The distinction between the theory of the symbolic equation ^ = 1, and that of 
the ordinary equation a::* — 1 = 0, presents itself in the very simplest case, n = 4. For, 
consider the group 

which are a system of roots of the symbolic equation 

There is, it is clear, at least one root /3, such that /S" = 1 ; we may therefore 
represent the group thus, 

1, a, /9, al3, (i8» = l); 

then multiplying each term by a as further factor, we have for the group 1, o*, afi, 
a'jS, so that a' must be equal either to or else to 1. In the former case the 
group is 

1, a, a\ a», (a*=l), 

which is analogous to the system of roots of the ordinary equation ic* — 1 = 0. For 
the sake of comparison with what follows, I remark, that, representing the last- 
mentioned group by 

1, a, A 7» 
we have the table 



1, 



a, 



A 



13 



1 


a 


13 
7 


7 


a 

13 


13 


1 


7 


1 


a 
/3 


7 


1 


a 



126 



ON THE THEOllY OF GEOUP8, 



[125 



If, on the other hand, ct* = l, then it is easy by similar reasoning to show that we 
must have a^ = /3cc, so that the group in the case is 

1, a, /9, a^e, (a« = l, /3» = 1, a/8 = /3a); 

or .if we represent the group by 

we have the table 

1 a i8 7 



/9 



1 


a 


/9 


7 


a 


1 


7 








7 


1 


a 

i 

1 


7 


/9 


a 



or, if we please, the symbols are such that 

a« = /3» = 7» = 1, 

/9 = 7« = «A 
7 =a/9 = i8a; 

[and we have thus a group essentially distinct from that of the system of roots of 
the ordinary equation ic* — 1 = 0]. 

Systems of this form are of frequent occurrence in analysis, and it is only on 
account of their extreme simplicity that they have not been expressly remarked. For 
instance, in the theory of elliptic functions, if n be the parameter, and 



/ \ ^ Q/ \ ^ + ^ / \ c»(l+n) 



then a, /8, y form a group of the species in question. So in the theory of quadratic 
forms, if 

a (a, 6, c) = (c, 6, a) 

0(a, 6, c) = (a, -6, c) 

7 (a, 6, c) = (c, - 6, a) ; 

although, indeed, in this case (treating forms which are properly equivalent as identical) 
we have a = /8, and therefore 7=1, in which point of view the group is simply a 
group of two symbols 1, «,(«'=!). 



125] AS DEPENDING ON THE SYMBOLIC EQUATION ^ = 1. 127 

Again, in the theory of matrices, if I denote the operation of inversion, and tr 
that of transposition, (I do not stop to explain the terms as the example may be 
passed over), we may write 

a = 7, i8 = tr, 7 = 7 . tr = tr . 7. 

I proceed to the case of a group of six symbols, 

1, a, )8, 7, 8, e, 
which may be considered as representing a system of roots of- the symbolic equation 

It is in the first place to be shown that there is at least one root which is a 
prime root of ^ = 1, or (to use a simpler expression) a root having the index 3. It 
is clear that if there were a prime root, or root having the index 6, the square of 
this root would have the index 3, it is therefore only necessary to show that it is 
impossible that all the roots should have the index 2. This may be done by means 
of a theorem which I shall for the present assume, viz. that if among the roots of 
the symbolic equation ^ = 1, there are contained a system of roots of the symbolic 
equation 6^=1 (or, in other words, if among the symbols forming a group of the order 
there are contained symbols forming a group of the order j>), then j> is a submultiple 
of n. In the particular case in question, a group of the order 4 cannot form part 
of the group of the order 6. Suppose, then, that 7, h are two roots of ^=1, having 
each of them the index 2; then if 7S had also the index 2, we should have 7S = S7; 
and 1, 7, 8, S7, which is part of the group of the order 6, would be a group of 
the order 4. It is easy to see that 7S must have the index 3, and that the group 
is, in fact, 1, 78, S7, 7, 8, 7S7, which is, in fact, one of the groups to be presently 
obtained; I prefer commencing with the assumption of a root having the index 8. 
Suppose that a is such a root, the group must clearly be of the form 

1, o, o*, 7, a7, a«7, (a»=l); 

and multipljring the entire group by 7 as nearer factor, it becomes 7, 07, a*7, 7-, 
ay, fltV; we must therefore have 7^ = 1, a, or a". But the supposition 7* = a' gives 
y* = a* = a, and the group is in this case 1, 7, 7*, 7*, 7*, 7" (7^=1); and the suppo- 
sition 7* = a gives also this same group. It only remains, therefore, to assume 7* = 1 ; 
ihen we must have either 7a = a7 or else 70 = 0*7. The former assumption leads to 
the group 

1, a, a", 7, 07, a^ (a» = l, 7*= I, l^^^l\ 

which is, in fact, analogous to the system of roots of the ordinary equation afi—\ = ^\ 
and by putting 07 = X, might be exhibited in the form 1, \, V, X', \*, V, (\« = 1), 
under which this system has previously been considered. The latter assumption leads 
to the group 

1, o, a«, 7, 07, o^ (a» = l, 7^ = 1, 7a = «^)> 

and we have thus two, and only two, essentially distinct forms of a group of six. 
If we represent the first of these two forms, viz. the group 

1, «, a*, 7» «% a'Vi (a' = l, 7' = 1> 7a=«7) 



128 



ON THE THEORY OF OBOUPS, 



[125 



by the general symbols 



we have the table 



1. a. /9, 7, S, e, 



1. a. /9, 



S, 







1 


a 


1 


7 


S 


t* 


a 


fi 


7 


8 


e 


1 


/9 


7 


S 


e 


1 


a 


7 


B 


e 


1 


a 

/9 


ys 


b 


€ 


1 


a 


7 


€ 


1 


a 


P 


7 


S 



while if we represent the second of these two forms, viz. the group 

1, a, a\ 7, 07, o^ (ct* = l, 7»=1, 70 = 0*7), 

by the same general symbols 

1, a, /9, 7» S» €, 
we have the table 

10/9786 



1 



/8 



1 


a 


ys 


7 


S 


e 


a 


/9 


1 


e 


7 


S 


y3 


1 


a 


S 


€ 


7 


7 


£ 


e 


1 





iS 


S 


€ 


7 


^ 


1 


a 


e 


7 


h 





1 



125] AS DEPENDING ON THE SYMBOLIC EQUATION $*'=l. 129 

or, what is the same thing, the system of equations is 

1 = ^a = 5r/9 = 7» =55=6*, 
a = /8^ = Sy = eS = ye, 
/8 = a' = €7 = yS = Se, 
fy = Set = e^ = ^g = a€, 

S = ea = y/8 = ay = /8e, 
£ = fya = S/8 = /9y = aS. 

An instance of a group of this kind is given by the permutation of three letters; 
the group 

1, a, 0f % ^» f 
may represent a group of substitutions as follows : — 

abc, cahy bca, acb, cba, bac 
abc abc abc abc abc abc. 

Another singular instance is given by the optical theorem proved in my paper 
"On a property of the Caustic by refraction of a Circle, [124]." 

It is, I think, worth noticing, that if, instead of considering a, /3, &c. as symbols 
of operation, we consider them as quantities (or, to use a more abstract term, *cogi- 
tables') such as the quaternion imaginaries; the equations expressing the existence 
of the group are, in fact, the equations defining the meaning of the product of two 
complex quantities of the form 

w + aa + 6/8 4- . . . ; 

thus, in the system just considered, 

(wH-aa + 6/9 + cy + dS + C6)(w;' + a'a + 6'/8 + cy + d'S + e'6)= W + Aa + B/S + Cy + DS + Ee, 

where 

W = tuw' + a6' + a'6 + cc' + dd' H- ee\ 

A =wa' + iji/a+ bV + dc' + edf + cc', 

5 = w6' + w/6 4- aa'-\- ed + cd' + de\ 

C =^wc' + w/c 4- da' + eV + 6d' + aefy 

D = wd! +v/d-{-ea' + cb' + ac' + be\ 

E =we' + w'e H- ca' + d6' + 6c' -^adf. 

It does not appear that there is in this system anjrthing analogous to the 
Baodulus <i;* + a^ + y* + -?', so important in the theory of quatemiona 

I hope shortly to resume the subject of the present paper, which is closely 
connected, not only with the theory of algebraical equations, but also with that of 

c. n. 17 



130 ON THE THEORY OF GROUPS, &C. [125 

the composition of quadratic forms, and the 'irregularity' in certain cases of the 
determinants of these forms. But I conclude for the present with the following two 
examples of groups of higher orders. The first of these is a group of eighteen, viz. 

1, a, A 7, ayS, i8a, 07, 70, /97, 7/8, a/87, /97a, 70/8, a^a, ^87^ 70:7, a^rtP, ^7^a, 

where 

ct« = l, /9» = 1, 7» = 1, (/87)» = 1, (7«)' = 1, («^)' = 1, (a/87)» = l, (/37a)» = l, (7«/8)» = l; 

and the other a group of twenty-seven, viz. 

1, a, Qi*, 7» y* 7«, «% 7«'» aY t"** ^T^i 7^. «V» 

07a, 07*a, 0*70:, 0*7*0, 070*, 07*0*, o^ct*, o^a", 707', 70*7*, 7*07, 7*0*7, 7*070*, 707*0*, 

where 

o» = l, 7»=1, (7o)' = l, (yo)» = l, (7Qt«)' = l, (7'c^)'=l. 

It is hardly necessary to remark, that each of these groups is in reality perfectly 
symmetric, the omitted terms being, in virtue of the equations defining the nature 
of the symbols, identical with some of the terms of the group: thus, in the group 
of 18, the equations a? = l, /9* = 1, 7^ = 1, (0/87)* = 1 give 0^7 = 7/80, and similarly for 
all the other omitted term& It is easy to see that in the group of 18 the index 
of each term is 2 or else 3, while in the group of 27 the index of each term is 3. 

2 SUyns Buildings, Nov. 2, 1853. 



126] 



131 



126. 



ON THE THEOKY OF GKOUPS, AS DEPENDING ON THE 
SYMBOLIC EQUATION <9»=1.— Second Part. 



[From the PkUoaophical Magazine, vol. vii. (1854), pp. 408 — 409.] 



Imagine the symboU 

L, M, N, ... 

Bach that (JL being any symbol of the system), 
w the group 

1, a, A---; 

then, in the first place, M being any other 83rmbol of the system, M^^L, M^^M, 

^''%,.. will be the same group 1, a, /9, In fact, the system Z, M, N,.., may be 

^tten Z, Za, L8...; and if e.g. if = Za, iV'=Z/9 then 

M-^N = (La)-' Z/9 = a-* Z"^ Z/9 = a-% 

which belongs to the group 1, a, A .... 

Next it may be shown that 

LL'\ ML-\ NL-\... 

^ & group, although not in general the same group as 1, a, ^, .... In bet, writing 
-^^ia, N=Lfi, &c., the i^mbols just written down are 

LL''\ LolL-\ Z/9Z-S... 

*^d we have e.g. LoJr' . L/SL"' = LajSL"' = LyL''\ where 7 belongs to the group I, a, 0, 

17—2 



132 ON THE THEORY OF GROUPS, &C. [126 

The system Z, M, JV, ... may be termed a group-holding system, or simply a 
holder; and with reference to the two groups to which it gives rise, may be said 
to hold on the nearer side the group Z~*Z, L~^M, Z""W, ..., and to hold on the 
further side the group ZZ~S LM"^, LN~^,.., Suppose that these groups are one and 
the same group 1, a, 13... y the system Z, Jf, -JT, ... is in this case termed a sym- 
metrical holder, and in reference to the last-mentioned group is said to hold such 
group symmetrically. It is evident that the symmetrical holder Z, M, N, ... may be 
expressed indiflferently and at pleasure in either of the two forms Z, Zo, Lj3,... and 
Z, ctZ, I3L', Le. we may say that the group is convertible with any symbol Z of 
the holder, and that the group operating upon, or operated upon by, a symbol Z of 
the holder, produces the holder. We may also say that the holder operated upon by, 
or operating upon, a symbol a of the group reproduces the holder. 

Suppose now that the group 

li a, A y> 8, €, f. ••• 
can be divided into a series of symmetrical holders of the smaller group 

1, a, ^, ... ; 

the former group is said to be a multiple of the latter group, and the latter group 
to be a submultiple of the former group. Thus considering the two different forms 
of a group of six, and first the form 

1, a, «•, 7» 7«» 7a'> (a* = l, 7* = !. «7=7a), 

the group of six is a multiple of the group of three, 1, a, o* (in fiwt, 1, a, o* 
and 7, 70, 7a* are each of them a symmetrical holder of the group 1, a, a'); and 
so in like manner the group of six is a multiple of the group of two, 1, 7 (in fact, 
1, 7 and a, a7, and a, 0*7 are each a symmetrical holder of the group 1, 7). There 
would not, in a case such as the one in question, be any harm in speaking of the 
group of six as the product of the two groups 1, a, a' and 1, 7, but upon the whole 
it is, I think, better to dispense with the expression. 

Considering, secondly, the other form of a group of six, viz. 

1, a, <^f 7» 7a, 7a"(a* = l, 7' = 1> a7 = 7a'); 

here the group of six is a multiple of the group of three, 1, a, a* (in fact, as be- 
fore, 1, a, a' and 7, 70, 70', are each a symmetrical holder of the group 1, o, o', 
since, as regards 7, 7a, 7a", we have (7, 70, 70') = 7(1, o, o«) = (l, o", 0)7). But 
the group of six is not a multiple of any group of two whatever; in feet, besides 
the group 1, 7 itself, there is not any symmetrical holder of this group 1, 7; and 
so, in like manner, with respect to the other groups of two, 1, 7a, and 1, 7a'. The 
group of three, 1, a, a\ is therefore, in the present case, the only submultiple of 
the group of six. 

It may be remarked, that if there be any number of symmetrical holdera of the 
same group, 1, a, )8, ... then any one of these holders bears to the aggregate of the 
holders a relation such as the submultiple of a group bears to such group; it is 
proper to notice that the aggregate of the holders is not of necessity itself a holder. 



127] 



133 



127. 



ON THE HOMOGRAPHIC TRANSFORMATION OF A SURFACE 

OF THE SECOND ORDER INTO ITSELF. 

[From the Philosophical Magazine, vol. vil (1854), pp. 208 — 212: continuation of 122.] 

I PASS to the improper transformation. Sir W. K Hamilton has given (in the note, 
p 723 of his Lectures on Quaternions [Dublin, 1853)] the following theorem : — If there 
l>e a polygon of 2m sides inscribed in a surface of the second order, and (2m - 1) of 
the sides pass through given points, then will the 2m-th side constantly touch two 
cones circumscribed about the surface of the second order. The relation between the 
extremities of the 2m-th side is that of two points connected by the general improper 
^fansformation ; in other words, if there be on a surface of the second order two 
points such that the line joining them touches two cones circumscribed about the 
stir&ce of the second order, then the two points are as regards the transformation 
^ question a pair of corresponding points, or simply a pair. But the relation between 
^he two points of a pair may be expressed in a different and much more simple 
form. For greater clearness call the surface of the second order Z7, and the sections 
along which it is touched by the two cones, 0, <(>; the cones themselves may, it is 
<^W, be spoken of as the cones 0, <f>. And let the two points be P, Q. The line 
•PQ touches the two cones, it is therefore the line of intersection of the tangent 
plane through P to the cone 0, and the tangent plane through P to the cone 0. 
l^t one of the generating lines through P meet the section in the point A, and 
the other of the generating lines through P meet the section ^ in the point B. 
The tangent planes through P to the cones 0, if> respectively are nothing else than 
the tangent planes to the surface U at the points A, B respectively. We have there- 
fore at these points two generating lines meeting in the point P; the other two 



134 ON THE HOMOGRAPHIC TRANSFORMATION OF A SURFACE [127 

generating lines at the points A, B meet in like manner in the point Q. Thus P, 
Q are opposite angles of a skew quadrangle formed by four generating lines (or, what 
is the same thing, Ijdng upon the surface of the second order), and having its other 
two angles, one of them on the section and the other on the section ^ ; and if 
we consider the side PA as belonging determinately to one or the other of the two 
systems of generating lines, then when P is given, the corresponding point Q is, it 
is clear, completely determined. What precedes may be recapitulated in the statement, 
that in the improper transformation of a surface of the second order into itself, we 
have, as corresponding points, the opposite angles of a skew quadrangle lying upon 
the surface, and having the other two opposite angles upon given plane sections of 
the surface. I may add, that attending only to the sections through the points of 
intersection of 0, ^, if the point P be situate anywhere in one of these sections, 
the point Q will be always situate in the other of these sections, Le. the sections 
correspond to each other in pairs; in particular, the sections 0, <f> are corresponding 
sections, so also are the sections 6, <I> (each of them two generating lines) made by 
tangent planes of the surface. Any three pairs of sections form an involution ; the 
two sections which are the sibiconjugates of the involution are of course such, that, 
if the point P be situate in either of these sections, the corresponding point Q will 
be situate in the same section. It may be noticed that when the two sections 0, <f> 
coincide, the line joining the corresponding points passes through a fixed point, viz. 
the pole of the plane of the coincident sections; in fact the lines PQ and AB are 
in every case reciprocal polars, and in the present case the line AB lies in a fixed 
plane, viz. the plane of the coincident sections, the line PQ passes therefore through 
the pole of this plane. This agrees with the remarks made in the first part of the 
present paper. 

The analytical investigation in the case where the sur&ce of the second order 
is represented under the form xy — zw = is so simple, that it is, I think, w^orth 
while to reproduce it here, although for several reasons I prefer exhibiting the final 
result in relation to the form ic" + y* + ;?* + t(;* = of the equation of the surface of 
the second order. I consider then the surface xy--zw = 0, and I take (a, /9, 7, S), 
(a\ /S', y, S') for the coordinates of the poles of the two sections 0, ^, and also 
(^i> yi> ^i> ^i)> (^> ya» ^a» ^«) fts the coordinates of the points P, Q. We have of course 
a^i^i — j^iWi = 0, flPi^a — jSjWj = 0. The generating lines through P are obtained by com- 
bining the equation xy — zw = of the surface with the equation wi/i + yxi^zwi^wzi^O 
of the tangent plane at P. Eliminating a firom these equations, and replacing in the 

result Xi by its value -^-^, we have the equation 

(y^ - -^yi) (ywi - t(;yi) = 0. 

We may if we please take yzi — zyi^O, asy^^-yxi-- zWi^-wzi^O as the equations of 
the line PA ; this leads to 

yzi-zyi^O, I yw;«-w;y« = 0, | 

«y 1 + y ^ "" -8^1 "" ^^1 = , j wy^ + y^a — ^«^j — ^^^ = o, j 



^a - zy^ = 0, I 

h + y^a — ^^a — t^^a = 0, j 



127] OF THE SECOND ORDER INTO ITSELF. 135 

for the equations of the lines FA^ QA respectively; and we have therefore the 
coordinates of. the point A^ coordinates which must satisfy the equation 

of the plane 0. This gives rise to the equation 

ya (ayi - S^i) - -J^a (7^1 - )9^i) = 0- 
We have in like manner 

ayi-¥yx^''ZWi-wzj = 0,] xy^ 

for the equations of the lines P£, QB respectively ; and we may thence find the 
coordinates of the point 5, coordinates which must satisfy the equation 

^x + a'y - h'z - 7't(; = 

of the plane ^. This gives rise to the equation 

y% («Vi - 7^1) - z^ (8^1 - /S'wi)- 

It is easy, by means of these two equations and the equation x^^ — z^w^^O, to form 
the system 

a?a = (ayi - S^i) (a'yi - 7^i)> 

ya == (7yi - fiz^) i^'yi - ^^i)» 

w/a = (ayi - S-?,) (S'yi - fi'^^i) ; 

or, effecting the multiplications and replacing z^^Wx by x^y^y the values of x^y y,, 2^3, t^;. 
contain the common foctor yj, which may be rejected. Also introducing on the left- 
hand sides the common £Eu;tor MM\ where Jlf = a/9 — 78, M^ = a')8' — ^'i\ the equations 
become 

MM'x^ = 7'&c, + OLd'yi — a'S^^i — a7'«!i, 

MMy^ = /9i8'^+ 7S'yi - P^z^ - i8'7«'i, 
MMz^ = )87'iri + 7a'yi — /Sa'^Tj — 77'Wi , 
MMw^ = fflxx + aS'yi - SS'^j - dfi'w^, 

^lues which give identically x^^ — z^^^ x^^ — z^w^. Moreover, by forming the value 
of the determinant, it is easy to verify that the transformation is in fact an im- 
proper, ona We have thus obtained the equations for the improper transformation of 
the surface xy^zw^^O into itself. By writing Xi + iyi, Xi^iyi for Xi, yi, &c., we have 
the following system of equations, in which (a, 6, c, d), (a\ b', &, d/) represent, as 
before, the coordinates of the poles of the plane sections, and Jlf' = a*H-5* + c' + d*, 
Jf'^«a'*H-6'«H-c'*H-(i^ viz. the system^ 

^ The system b very similar in form to, but is euentially different from, that which conld be obtained 
^ the theory of qnatemionB by writing 

the lasfc-mentioned transformatiGn is, in &ot, j^roper, and not improper. 



136 ON THE HOMOGRAPHIC TRANSFORMATION OF A SURFACE [127 

MWx^ = (oa' - hV - cc' - ddf) x^ + { ah' + a'b + cd' - c'd) yi 

+ ( oc' + a'c + d6' - d'h) -g:, + ( ad'+ a'd + he - 6'c) w^, 
iOfy, = (a6' + a'6- cd' + c'd) iCi + (- oa' + 66' - cc/ - dd')y^ 

+ ( 6c' + 6'c - da' + d'a) ^x + ( 6d' + 6'd - oc' + a'c) w, 
MM'z^ = (ac' + a'c - d6' + d'6) a^ + ( 6c' + 6'c '-'ad'+ a'd) yi 

-{-(-aa'- 66' + cc' - dd') ;?i + ( cd' -{- dd - 6a' + 6'a) w,, 
iOf'm, = (ad' + a'd- 6c' + 6'c) (x\-\-{ 6d' + 6'd -ca '\' da) y^ 

+ ( cd' -{'dd- a6'+ a'6) z^ + (- aa' - 66' ''cd + dd') w^, 

values which of course satisfy identically x^-\-y^-\-z^-\-w^ = x^-\-y^-\-z^ + w^y and which 
belong to an improper transformation. We have thus obtained the improper trans- 
formation of the surface of the second order a^ + y' + 2:* + t(;* = into itself 

Returning for a moment to the equations which belong to the surface xy'^zw = 0, 
it is easy, to see that we may without loss of generality write a = ^=a' = )8'=0; 
the equations take then the very simple form 

MWx^^yBx,, MWy^^yB'yu M]irz^ = -yy'w,, itf Jf' w, = - SS'^j, 

where MJiT = V — 78 V — 7'S' ; and it thus becomes very easy to verify the geometrical 
interpretation of the formulae. 

It is necessary to remark, that, whenever the coordinates of the points Q are 
connected with the coordinates of the points B by means of the equations which 
belong to an improper transformation, the points P, Q have to each other the 
geometrical relation above mentioned, viz. there exist two plane sections 0, <l> such 
that P, Q are the opposite angles of a skew quadrangle upon the surface, and having 
the other two opposite angles in the sections 0, <(> respectively. Hence combining 
the theory with that of the proper transformation, we see that if A and B, B and 
C, ..., M and N are points corresponding to each other properly or improperly, then will 
JV and A be points corresponding to each other, viz. properly or improperly, according 
as the number of the improper pairs in the series A and B, B and C, ..., M and N 
is even or odd; i.e. if all the sides but one of a polygon satisfy the geometrical 
conditions in virtue of which their extremities are pairs of corresponding points, the 
remaining side will satisfy the geometrical condition in virtue of which its extremities 
will be a pair of corresponding points, the pair being proper or improper according 
to the rule just explained. 

I conclude with the remark, that we may by means of two plane sections of a 
surface of the second order obtain a proper transformation. For, if the generating 
lines through P meet the sections 0, <(> in the points A, B respectively, and the 
remaining generating lines through A, B respectively meet the sections ^, respec- 
tively in R, A', and the remaining generating lines through P', A' respectively meet 
in a point P'; then will P, P' be a pair of corresponding points in a proper trans- 



127] OF THE SECOND ORDER INTO ITSELF, 137 

formation. In fact, the generating lines through P meeting the sections 0, <f> m the 
points Af B respectively, and the remaining generating lines through A, B respectively 
meeting as before in the point Q, then P and Q will correspond to each other im- 
properly, and in like manner R and Q will correspond to each other improperly; i.e. 
P and P* will correspond to each other properly. The relation between P, P' may 
be expressed by saying that these points are opposite angles of the skew hexagon 
PARP'A'B lying upon the surface, and having the opposite angles A, A' in the 
section 0, and the opposite angles B, R in the section <(>. It is, however, clear from 
what precedes, that the points P, P' lie in a section passing through the points of 
intersection of 0, ^, and thus the proper transformation so obtained is not the general 
proper transformation. 

2 Stone Buildings, January 11, 1854. 



c. n. 



18 



138 



[128 



128. 



DEVELOPMENTS ON THE PORISM OF THE IN-AND-CIRCUM- 

SCRIBED POLYGON. 



[From the Philosophical Magazine, vol. viL (1854), pp. 339 — 345.] 

I PROPOSE to develops some particular cases of the theorems given in my 
paper, "Correction of two Theorems relating to the Porism of the in-and-circumscribed 
Polygon" {Phil. Mag, voL vi. (1853), [116]). The two theorems are as follows: 

Theorem. The condition that there may be inscribed in the conic [7=0 an 
infinity of n-gons circumscribed about the conic F=0, depends upon the development 
in ascending powers of f of the square root of the discriminant of ^U+V\ viz. if 
this square root be 



then for n = 3, 5, 7, &c. respectively, the conditions are 



|C|=0, 



C, D =0, 

D, E 



G, D, E 

D. E. F 

E, F, 



= 0, &c ; 



and for n = 4, 6, 8, &c. respectively, the conditions are 



D|=0, 



D, E =0, 

E, F I 



D, E, F 

E, F, G 

F, G, H 



= 0, &c. 



128] DEVELOPMENTS ON THE PORISM, &0. 139 

Theorem. In the case where the conies are replaced by the two circles 

then the discriminant, the square root of which gives the series 

il + £f + Cp + Df + ^f * + &c. , 

is 

Write for a moment 

^+£f+Cp + i)f> + ^f* + &c. = V(l + af)(l+6f)(l + cf), 

then 

^ = 1. 

25 = a + 6 + c, 
- 8C = a» + 6^ + c«- 26c - 2ca - 2a6, 
162) = a» + 6» + c'-a* (6 +c)- 6^(0 + a)- c»(a +6) + 2a6c, 
-128i; = 5a* + 56* + 5c*-4a»(6 + c)-46»(c+a)-4c»(a + 6) 

+ 4a»6c + 46»ca + 4c«a6 - 26»c» - 2c»a» - 2a>6«, 
&c. 

To adapt these to the case of the two circles, we have to write 

r-(l + af)(l + 6f)(l + cf) = (l+f){r»+f(r>+i?-a«) + f'i?l, 

and therefore 

c = l, 



values which after some reductions give 



^=1. 

-r«.8C = (i? - a')" - 4i2V, 
r* . 162) = (i? - o») {(i? - o')' - 2r» (i? + a% 
-r* . 128^ = 5 (i? -o«)« - 8 (iJ»-o»)'(i? + 2r>)r»+ 16a«r«. 



Hence also 



»* . 1024 (GE - i)*) = {5 (i? - o«)* - 8 (i? - o»)' (ii' + 2r») r» + 16aV} {(i? - o>)* - 4i2V)} 

- 4 {(i? - o>)' - 2 (ii» - o») (i? + a') r»j', 

18—2 



140 DEVELOPMENTS ON THE POMSM OF THE [128 

which after all reductions is 

+ 16J? (i? + 2a») (i? - a»)» r* 

Hence the condition that there may be, inscribed in the circle a;' + y' — i? = 
and circumscribed about the circle' (a? — a)" + y* — r* = 0, an infinity of n-gons, is for 
n = 3, 4, 5, Le. in the case of a triangle, a quadrangle and a pentagon respectively, 
as follows. 

I. For the triangle, the relation is 

(i? - a»)» - 4iJV* = 0, 

which is the completely rationalized form (the simple power of a radius being of 
course analytically a radical) of the well-known equation 

which expresses the relation between the radii R, r of the circumscribed and inscribed 
circles, and the distance a between their centres. 

II. For the quadrangle, the relation is 

(i?-a»)»-2r»(J? + a») = 0, 
which may also be written 

(i2+r + a)(iJ + r-tt;(i2-r + a)(-B-r-a)-r* = 0. 
(Steiner, Crelle, t. ii. [1827] p. 289.) 

III. For the pentagon, the relation is 

(R''-'a''y - 12i? (i? -tt»)*r» + 16i?(i? + 2A^) (iP-a»)"r* - 64i?tt*r« = 0, 
which may also be written 

(iP - a^y {(i? - a^y - 4iPr»}» - 4iPr» {(iP - a^y - 4a»r»}» = 0. 

The equation may therefore be considered as the completely rationalized form of 

(i?-a*)> + 2iJ(i?-a»)»r-4i?(iP-a»)r»-8i2a»r«=a 

This is, in fact, the form given by Fuss in his memoir "De polygonis symme- 
trice irregularibus circulo simul inscriptis et circumscriptis," Nova Acta Petrop. t. xiii. 
[1802] pp. 166 — 189 (I quote from Jacobi's memoir, to be presently referred to). Fuss 
puts iJ + a=jt>, R — a = qy and he finds the equation 



jj^q^ - r' (/)» + y*) _ / g - r 



128] IN-AND-CIRCUMSCRIBED POLYGON. 141 

which, he remarks, is satisfied by r^^—p and r= -^ , and that consequently the 

rationalized equation will divide by p + r and pq — r{p-{-q); and he finds, after the 
division, 

P^^+p^^ip + q)r-pq(p + qyf^'-(p-\-q)(P'-qyr'=^0, 
which, restoring for p, q their values R + a, R — a, is the very equation above found. 

The form given by Steiner (CreUe, t. ii. p. 289) is 

r (iZ - a) = (iZ + a) V(i2 -T^M^) ~(^^r;^"^) + (22 + a^ 

which, putting p, q instead of iJ + a, -B — a, is 

qr =p 'Jip -r){q-r)-{-p V(g - r) (g +p) ; 

and Jacobi has shown in his memoir, "Anwendung der elliptischen Transcendenten 
u. 8. w.," CreUey t. III. [1828] p. 376, that the rationalized equation divides (like that 
of Fuss) by the factor pq-'ip + q)r, and becomes by that means identical with the 
rational equation given by Fuss. 

In the case of two concentric circles a = 0, and putting for greater simplicity 
. = Jlf, we have 

il+£f + Cp + Df> + ^f* + &c. = (l + f)Vl + Jff. 

This is, in fact, the very formula which corresponds to the general case of two 
conies having double contact For suppose that the polygon is inscribed in the conic 
(7=0, and circumscribed about the conic C7' + P* = 0, we have then to find the 
discriminant of (U+U + P*, Le. of (l + f)Cr+P*. Let K be the discriminant of U, 
and let F be what the polar reciprocal of U becomes when the variables are replaced 
by the coefficients of P, or, what is the same thing, let — ^ be the determinant 
obtained by bordering K (considered as a matrix) with the coefficients of P. The 
discriminant of (l+f)i7+P" is (1 + f)»ir + (l + f)»P, Le. it is 

(l+f)>{J5r(l+f) + P}, =(J5r + P)(l + f)>(l + iff), 

K 

^here M== j^ — jj,; or, what is the same thing, M is the discriminant of U divided 

by the discriminant of U-^P*, And M having this meaning, the condition of there 
being inscribed in the conic (7 = an infinity of n-gons circumscribed about the conic 
^ + P* = 0, is found by means of the series 

^+5f+6'f»+i)p + ^f* + &c. = (l + f)Vl+iff. 



We have, therefore, 



DEVELOPMENTS ON THE P0RI8M OF THE 
A=l, 

16/) = JP-2Jtf', 
~12HE = 5M*-8M', 

1024(C£-D') = i£*(AP-12if+16), 

Hence for the triangle, quadrangle and pentagon, the conditions are — 
I. For the triangle. 



[128 



IL For the quadrangle, 
III. For the pentagon. 



if + 2 = 0. 
Jf- 4 = 0. 



and so on. 

It is worth noticing, that, in the case of two conica having a i-point contact, 
we have F = 0, and consequently M=l. The discriminant is therefore (l+f)*, and 
as this does not contain any variable parameter, the conies cannot be determined so 
that there may be for a given value of « (nor, indeed, for any value whatever of 
n) an infinity of n-gons inscribed in the one conic, and circumscribed about the 
other conic. 

The geometrical properties of a triangle, Sk. inscribed in a conic and circum- 
scribed about another conic, these two conies having double contact with each other. 




are at once obtained from those of the system in which the two conies are replaced 



128] IN-AND-CIRCUMSCBIBED POLYGON. 143 

by concentric circlea Thus, in the case of a triangle, if ABC be the triangle, and 
a, j8, 7 be the points of contact of the eidee with the inscribed conic, then the tangents 
to the circumscribed conic at A, B, C meet the opposite sides BC, CA, AB in points 
lying in the chord of contact, the linea Aa, Bff, Cy meet in the pole of contact, 
and BO on. 

In the case of a quadrangle, if ACEQ he the quadrangle, and b, d, f, h the 
pointa of contact with the inscribed conic, theu the tangents to the circumscribed 




conic at the pair of opposite angles A, E and the corresponding diagonal CQ, and 
in like manner the tangents at the pair of opposite angles' C, Q and the corresponding 
diagonal AB, meet in the chord of contact. Again, the pairs of opposite sides AC, 
£6, and the line dk joining the points of contact of the other two sides with the 
inscribed conic, and the pairs of opposite sides AO, GE, and the line hf joining the 
points of contact of the other two sides with the inscribed conic, meet in the chord 
of contact The diagonals AB, CQ, and the lines hf, dh through the points of 
coDtact of pairs of opposite sides with the inscribed conic, meet in the pole of 
^tact, &c 

The beautiihl systems of 'focal relations' for regular polygons (in particular for 
tlw pentagon and the hexagon), given in Sir W, B. Hamilton's Lectures on Quaternions, 
[Dublin, 1853] Noa. 379 — 393, belong, it is clear, to polygons which are inscribed in and 
tuwmscribed about two conies having double contact with each other. In foot, the focus 
of s conic is a point such that the lines joining such point with the circular points at 
"ifinity (Le, the points in which a circle is intersected by the line infinity) are tangents 
^ the conic In the case of two concentric circles, these are to be considered as 
''"Jelling in the circular points at infinity; and consequently, when the concentric 
cucles are replaced by two conies having double contact, the circular points at infinity 
*K replaced by the points of contact of the two conies. 



144 DEVELOPMENTS OK THE PORISM, &C. [128 

Thus, in the figure (which is simply Sir W, E. Hamilton's figure 81 put into 




perspectiveX the system of relations 

F. G{..)ABCI, 

G. H{..)BCDK, 
B,I(..)CDEF, 
I, K(..)DEAG, 
K, F(. .) EABH. 

will mean, F, 0(.,)ABCI, that there is a conic inscribed in the quadrilateral ABCI 
such that the tangents to this conic through the points F and pass two and two 
through the points of contact of the circumscribed and the inscribed conies, and 
similarly for the other relations of the system. As the figure is drawn, the tangents 
in question are of course (aa the tangents through the foci in the case of the two 
concentric circles) imaginary, 

2 Stone Buildivgs, March 7, 1854. 



129] 



145 



I 

[ 



129. 



ON THE PORISM OF THE IN-AND-CIRCUMSCRIBED TRIANGLE, 
AND ON AN IRRATIONAL TRANSFORMATION OF TWO TER- 
NARY QUADRATIC FORMS EACH INTO ITSELF. 

[From the Philosophical Magazine, vol. ix. (1855), pp. 513 — 517.] 

There is an irrational transformation of two ternary quadratic forms each into 
itself based upon the solution of the following geometrical problem, 

Given that the line 

Ix -\- my -\-nz = 

Dtteets the conic 

(a, 6, c, /, g, A$a?, y, zy = 

in the point (o^, y,, Zi); to find the other point of intersection. 

The solution is exceedingly simple. Take {a^, y^, z^) for the coordinates of the 
other point of intersection, we must have identically with respect to a?, y, z, 

(a, ...$a?, y, zy.(fSL, ...$i, m, wy-&(te + my + mr)» 

= (a,...$aH, yi, z^l^x, y, z).(a, ...Ja?,, y^, z^'^Xy y, z) 

^ a constant foctor pris. 

Assume successively a?, y, ^r = ®, |^, ffi ; |^, 98, Jp ; ffi, ip, Ct ; it follows that 



C. n. 



19 



146 ON THE PORISM OF THE IN-AND-CIRCUMSCRIBED TRIANGLE, [129 

or, what is the same thing, 

X.2 : y^ : Zt= yi^i (frw* + cfn^— 2/mn) 

' ^1^1 (ctw' + &n* — 2hlm), 

It is not necessary for the present purpose, but it may be as well to give the 
corresponding solution of the problem : 

Given that one of the tangents through the point (f , 17, f) to the conic 

(a, 6, c, /, g, A$a?, y, zf = 
is the line fia? -f 7?i,y + ni^ = ; to find the equation to the other tangent. 

Let Ijc + m^ + n^^ = be the other tangent, then 

(a, ...$f, 17, t;y.{a,...\x, y, zf -[{a ...J^, 17, t$a?, y, z)Y 

to a constant factor prh. Assume successively y = 0, ^ = 0; £^ = 0, a? = 0; a: = 0, y = 0; 
then we have 

i, : m, : 71,= miWi {a (a, . . . $f , 17, ?)"-(af + Ai7+5rJ)»} 

: n,Z,{6(a,...$f, 17, ?)» - (Af + 617 +y?yi 
: iim,{c(a,...$f, 17, r)»-((7f+/^ + cO'}; 
or, as they may be more simply written, 

i, : 771, : n,= 7/^1^ (J8(r* + ®i7« + 2JP17O 

: 71, fi (ODp + ia?» - 2® tf ) 

Returning now to the solution of the first problem, I shall for the sake of 
simplicity consider the formulae obtained by taking for the equation of the conic, 

«^ + /Sy* + 7-8^ = 0. 

We see, therefore, that if this conic be intersected by the line lx-\-my + nz=^Q in 
the points {x^, y,, z^ and {x^, y„ j?,), then 

«j : yj : -^a = yiZi (77?i» + a/«») 

: ZiX^{an^ -\-fifi) 



129] AND ON AN IRRATIONAL TRANSFORMATION &C. 147 

We have, in fact, identically 

lyi^i (^w* + 7W1") + mZiXi (yP + cm*) + rw^i (am* + 0P) 

= {amnxi + firdyi + ylmzi) {Ix^ + myi + nz^) — J?/m {ax^ + /9yi' + 7-2^1'). 

«yi V (/9»" + 7*^")" + fiziW (yl* + an*)« + 7^1 V («^' + ^^')" 

= a/97 { - PiTi' — m'yi* — n'^i* 

+ (twyi + nzi) Pa?i* + (n-2:, + Ix^) mh/^ + (Za^ + my^ n^y^ - Zlmnx^y^Zi] {Ix^ 4- wy, + /l^i) 

- (I'fiyxi" + w Vyi' + n^fiz,') (aci* + )9y,« + 7V) ; 

which show that if Ixi-^-myi^-nZi-O and aa?i» + )9yi* + 71^* = 0, then also lxi-^mya + nzi = 
and OiT,* + /Sya' + 7-er,* = : this is, of course, as it should be. 

I shall now consider Z, m, n as ffiven functions of Xi, yi, Zi satisfjdng identically the 
equations 

Ixi +myi +nzi =0, 

IHk + m^ca + n^b = 0, 

equations which express that lx + my + nz = is the tangent from the point (xi, y^, Zi) 
to the conic cur" + 6y* + cz^ = 0. And I shall take for a, /9, 7 the following values, viz. 

a = ax* + by I* + cz^* - a (a?i* 4- y^* + ^1*) , 
/9 = cuTi* + byi* + c-gTi* - 6 (x^* + y,* + ^1*), 
7 = cur,' + 6yi' + c^i' - c (iTi* + yi* + -2^1*); 

80 that iTi, yi, ^1 continuing absolutely indeterminate, we have identically ax^-k- ^yi^ + yz^ = ^. 
Also taking as a function of Xi, y^ ^,, the value of which will be subsequently 
pven, I write 

X2 = ©yi^i (fin* + 7m»), 

y2 = ^ZjXi(yP +an% 

Z2 = Sx^i(am*+l3t'); 

^ that Xi, yi, ^1 are arbitrary, and x^, y^, z^ are taken to be determinate functions 
^^ *i> yi, -2?!. The point (arj, yj, ^j) is geometrically connected with the point (xi, y^ z^) 
^ follows, viz. (a?a, yj, ^,) is the point in which the tangent through (xi, yi, z{) to 
the conic cur" + 6y* + c^* = meets the conic passing through the point (a^, yi, ^Tj) and 
tie points of intersection of the conies ax* + b^ + cz* = and a:* + y* + ^' = 0. Con- 
^uently, in the particular case in which (a?i, yi, ^1) is a point on the conic 
^+y'+^ = 0, the point (x^, ya, -^2) is the point in which this conic is met by the 
t^gent through (a^, y,, Zi) to the conic ax^ + by* + cz* = 0, 

It has already been seen that Ix^ + myi + n^i = and aa?i^ + /9yi* + 7^1' = identically ; 
consequently we have identically Ix^ 4- my^ + w^Tj = and ouv^* + fiy^^ + 7^2' = 0. The latter 
equation, written under the form 

(ax^* + by,* + cz*) {xf + y,' + z^") - (iCi» + yi» + ^1') (cw?,- + by.^ + c^aO = «, 

19—2 



148 ON THE PORISM OF THE IN-AND-CIRCUM80RIBED TRIANGLE, [129 

shows that if a^^ y„ z^ are such that a?j' + yj' + -?,• = a^' + y^' + ^j', then that also 
aoi^^ + by^ + cz^=^axi* + byi* + czi\ I proceed to determine O so that we may have 
^i* + yi' + V = ^' + yi' + V« We obtain immediately 

^,(^«' + y«' + '^>') = (^V + mV + »V)(aV + i8»yi« + 7«^i«) 

write for a moment 

CM?i* + 6yi' + C2r,'=p, iTi' + yi' + -2^1* = J, so that a=*p^aq, fi^p — bq, y^^p — cq, 
then 

o»ah' + I3h/,^ + T*^,' = 2p* - 2p .;)5 + (aV + 6»yi« + (fz^*) (f,^q {(aV + ^V + c»^i') 3 -i>*}. 

= J {(6 - c)« yiV + (c - a)» 2r,V + (a - 6)' fla'y,'}, 
a?iV + i8*wiV + Y*»iV - 2fiymWyi%* - 2yan^l^zW - 2a/8i»m'j?i«yi« 

— 2pg {oZ^ar/ + fcm^y/ + c?i*V — (6 + c) rnWy^z^ - {c + a) nH^z^x^ — (a + 6) 1hii^x^y^\ 

the first line of which vanishes in virtue of the equation Ix^ + myx + w^i = ; we have 
therefore 

^(a^'+y,' + -?,•) -H (^1* + yi» + z^^) 

= (Z V + ^V + wV) {(6 - c)» yi V + (c - a)» Vj?i* + (a - 6)» iCj V} 
+ 2 (aa?i* + 6yi' + cz^^) {aZV + &^V + cn%^-(b + c) mWy^z^- (c + a) nH%W - (a + 6) ?mV,*yi»} 

- (^' + yi' + -8^1*) {a'^V + 6*m^/ + c^V - 26cm«n«yi V - ican^l^z^x^ - iabl^Wyi^]- 
Hence reducing the function on the right-hand side, and putting 

we have 

+ (c*wi* - 26«m V) yi V + (cM - 2c»n*P) -^i V + ( W* - 2a«Pm«) aj^ V 

+ (bW - 2c*mV) yiV + (c»Z* - 2aWZ«) ^^ V + (aHn* - 26»Z>m«) iCj^y/ 

+ ZmW {bc-ca''ab) + 2n'Z' (-6c + ca-a6) + 2Pm* ( - 6c - ca + a6)} Xi%%\ 

The value of might probably be expressed in a more simple form by means 
of the equations Ixi + myi + n^i = and Z'6c + m^ca + n*a6 = 0, even without solving 
these equations; but this I shall not at present inquire into. 



129] AND ON AN IRRATIONAL TRANSFORMATION &C. 149 

Recapitulating^ I, m, n are considered as functions of ^, y^, ^i determined (to a 
common £EU^tor pria) by the equations 

i*a + wiyi +nzi =0, 
Pbc + rn^ca + n^ab = ; 

B is determined as above, and then writing 



we have 



y, = ©^iri(7p +an'), 
^, =5 ©a?iyi (am* + fiV) ; 



and these values give 



Is^ +my, + n2r, = 0, 

xf +y,' +-?,» =a?i« +yi' +zx\ 

axf + tyt* + ozf = cuTi' + 6yi' + c^i*. 

In connexion with the subject I may add the following transformation, viz. if 

Z^Jaaf^ V3/8 (y - -f) + V(3a - 2/8) (dj» + y» + z") + 2/8 (y;? + ;?a? + «y), 

f 

then reciprocally 

3>//8a?=r-V3a(y'-/) + V(3/8-2a)(a/» + y'» + /«) + 2a(y'/ + /«^ + fl?'yO. 

a^ + y» + ^ =a?'» + y'« + /«, 

/8 (aj* + y* + jt* -y-z^ - -^a? - icy) = a (a^» + y'* + /' -y'-^' - -2'a?' - a?'yO. 

Suppose 1 + /:> + p^ = 0, then 

a^ + j^-{-2^ — yz'-zx — xy = (x + py + p^z) (a? + p* + p^) ; 
and in feet 

3Va(a?' + py'+p»/) = ->/3)8(l + 2p)(a? + py + p«^), 

3Va(a?'+py+p^) = V3i8(l + 2p)(a? + p»y + p^). 
The preceding investigations have been in my possession for about eighteen months. 

2. Stone Buildings, April 18, 1855. 



150 



[130 



130. 



DEUXIEME MEMOIRE SUR LES FONCTIONS DOUBLEMENT 

PERIODIQUES. 



[From the Journal de Maihimatiques Pures et Appliques (Liouville), torn. xix. (1854), 

pp. 193—208: Sequel to Memoir t. x. (1845), 35.] 

Je vais essayer de d^velopper ici les propri^t^s qui se rapportent aux transformations 
lin^aires des p^riodes des fonctions yw, gx, Gx, Zx, dont je me suis occupy dans le 
M^moire sur les fonctions doublement p^riodiques que j*ai donn^ dans ce Recueil en 
1845. Avant d*entrer en matifere, je remai^que que partant des expressions 



des deux p^riodes, oi i = V — 1, on obtient, en ^crivant 

fl* = ck) — a)'i, 

T»=i;-i/i, 
les Equations 

ft*T = wu + G) V + i (W - w'u), 
au moyen desquelles et des valeurs 



IIT mod. {(OV - do'v) ' SIT mod. {mv - (o'u) 

des quantity fi, B, on ddduitles formules 



ft mod. {(OV — a)'i;) ' T mod. («t;' — (o'v) ' 



130] DEUXltlfB ICl^OIRE SUB LES FONCTIONS DOUBLEMENT PilBIODIQUES. 151 

Je ne &is attention qu'aux transformations qui correspondent k des entiers impairs et 
premiers, et je suppose, de plus, que la transformation soit toujours propre et r^gulifere ; 
c'est-i-dire qu'en ^rivant 

(2* + 1) ft, = xn + mT = (X, /Lt), 

(2A + l)T, = i;ft + pT = (i/, p), 

oh 2k +1 est un entier positif, impair et premier, et oh X, fi, v, p sont des entiers 
tels, qu'au signe pr^ \p-^fiv soit 6gsA k 2A; + 1, je suppose 

Xp — fiv=2k + 1, 

(condition pour que la transformation soit propre), et, en outre, 

X= 1, /iA = 0, (mod. 2) 
1^ = 0, p=l, 

(condition pour que la transformation soit r^guli^re). 

On trouve tout de suite 

ft= pn,-/tT, = (p, -/^X, 

T=-.|/n, + XT, = (-i., XX; 

j'^cris aussi 

ft, « tt), + ©/», ft,* = tt), - «/», 

et je suppose que i3„ /3, soient des fonctions de o>,, v, telles que les fonctions B, 0, 
de 0, V. 

Cela ^tant, je forme d'abord T^uation 

(2* + 1) (o),v/ - o)>,) = 0)1/' - w't;, 
au moyen de laquelle T^uation 

^ ^^ mod. (a)i/ - G)'i;) 
8e transforme en 

Beli 

i<"^')(^^^>-^-i"- „,.d(.;--.>.) {°--°^=nf^}' 



mod. 



mod. 






9rt(<»,v/-o)>,) 
~ nX mod («,v/ - «>,) ^"' "^ '* * "' • 



152 DEUXlklCB M^OIBE SUB LES FONCTIONS DOUBLEMENT P^RXODIQUES. [130 

oa enfin 



et de mSme 



{2k + l(B-fi)-B,]a^^fi^(p, X); 
Equations qui seront bientdt utiles. 

Je suppose d'abord que 2k + 1 soit ^gal k Tunit^, transformation que Ton peut 
nommer trivicUe, La fonction yx est d^finie par I'^quation 

ya? = e-**^a?n|l + ^^^J, mod.(m, n)< T, r= oo; 

dans (m, n) = mCl + nT, les entiers m, n doivent prendre toutes les valeurs positives 
ou negatives (le seul syst^me m = 0, n = except^) qui satisfont k Tin^galit^ 

mod. (m, n) < T, 

dont le second membre T sera ensuite suppose infini. Soit y,x la fonction corres- 
pondante pour les p^riodes fl^ , T^ ; on aura 

V ar = €-**.«• aril |l +7-^.1, mod.(m, ny<r, r=«. 
Or 

= m (Xil + fiT) + n (i/fl + pT), 
= (Xm + vn)il + (jjLfn + pn) T, 
= m^fl + n,T, 

En ^rivant, comme nous venons de le faire, 

m, = Xm + im, 

on voit tout do suite qu'ii chaque systfeme de valeurs entiferes de m, n, correspond 
un Hystfemo, ct un seul systfeme, de valeurs enti^res de m^, w,; et que de m6me k 
chaque syst^me de valeurs entiferes de m,, n,, correspond un systfeme, et un seul syst^me, 
do valeurs untiferes do m, w ; de plus, les systfemes m = 0, n = et m, = 0, n^ = 0, 
correspondent Tun k Tautre. II est done permis d'^rire 



( (m, n)j ( (m, n),) 



les limitoH comme auparavant ; car, k cause de 

(m, n), = (m„ n,% 



gx 



130] DEUXrfeMB Ml^OIRE SUR LES FONCTIONS DOUBLEMENT PiaaiODIQUBS. 153 

la condition pour les limites, savoir : 

mod (m, n\ < T, T= oo , 

devient 

mod. (m, n) < r, T = oo . 

Cela donne enfin I'^uation 

et, au moyen de cette Equation, on obtient une ^nation correspondante pour la trans- 
formation de Tune quelconque des fonctions yx^ gx, Ox, Zx, d^finies par les ^nations 

yar = 6-**^.a;n|l+^^^^|, mod. (m,n)<r, 

= e-**^. nil 4-^-^^}, mod. (m, n) < T, r=oo; 

Ox^e^^. n(l+7-^J, mod. (m, fi) < r, 

\ (m, n)j 

Za? = €-**^. u\l+'-J-X, mod. (m, n) < r, 

(^nations dans lesquelles m = m + ^, n = n + ^). Je prends par exemple la fonction gx, 
et j'&ris dans I'^uation entre y^x et yx, x + ^fl au lieu de x. Soit pour un moment 
p = 2p' 4- 1, /A = 2fi\ cela donne 

a?+jn = ar + i(/)ft,-A*T,)=a? + (p',-/). 

Done 

y, (x + ill) = e^.*'P. '*')' M^g^x, 
c'est-i-dire 

y, (a? + ifl) = 6*^'*<^» **>' if,g,aj ; 
de plus, 

y (x + ill) = 6*^0* ifgar. 

Ces substitutions ^tant eflFectu(?es, les coefficients M, M, doivent 6tre ^limin^s en ^rivant 
««0; cela donne 

ou enfin, au moyen d'une ^nation ddji trouv^, 
et de m^me pour les fonctions Ox, Zx, 

c. n. 20 



154 DEUXlilME ICl^OIRE SUB LES FONCTIONS DOUBLEMENT P^RIODIQUES. [130 

Done eofin, en reprdsentant par Jx Tune quelconque des fonctions jx, gx, Ox, Zx, 
on aura 

oil J,x est ce que devient Jx au moyen d'une transformation triviale (propre ct 
r^gulifere) des p^riodes. 

Je passe k pr^nt k la transformation pour un nombre impair et premier {2k + 1) 
quelconque; mais pour cela on a besoin de connattre la valeur de la fonction 

"' = "{^-^ (m, r)+y }' ^od.[(m.n) + y}<T, r=oc. 

o{i y = a + bi est une quantity r^elle ou imaginaire quelconque. 

Soit u ce que devient u' en prenant pour la condition par rapport aux limites 

mod. (rw, n) < T, 7= oo ; 
on trouve sans peine 

y(y) 

Pour trouver u\ je forme T^uation 

u : u' = u\l 4- . ^ J , 

la limite inf^rieure du produit infini double ^tant 

mod. {(m, n) + y]> T, 
et la limite sup^rieure 

mod. (m, n) < T, T= x ; 

cela donne 

logtt-log«'=..S(-^;i7^-i^S,(^, „\+y}.+ -. 

car on peut d^montrer que 

S 7-^=0. ^7-^=0, &c. 
■^^ (ot, n)* (m, n)* 

Pour cela, observons que m et n ^tant infinis puisque T Test, la premifere des sommes 
dont il s'agit peut se remplacer par rint^grale double 

J __ rCdmdn 



130] DSUXI^ME M]^OIB£ SUB LES FONCTIONS DOUBLEMENT PilBIODIQUES. 155 

laquelle (en Ajrivant m = r cos d, n = r sin ^, ce qui donne, comme on sait, dmdn = rdrdff) 
devient 



-// 



drd0 



r{ilcos0-{'r8ui0y' 
rfoti 

(log r) dd 



-II 



(aco&e+rQindy 

en prenant (logr) entre les limites convenablea Pour trouver ces limites, j'^ris 

(m, n) + y = r(ncos^ + TsiD^) + y; 

ce qui donne 

mod.' {(m, n) + y} = {r (flcos ^ + T sin ^) + y} {r(fl* cos d + T*8in 0) + y*}, 
fiavoir, k Tune des limites 

r»(ftcos^ + T8ind)(n*co8^ + T*8in^) 

+ r{y*(nco8^ + T8in^) + y(ft*cos^ + T*sin^)} + 2* = 0; 

ou, en n^gligeant les puissances negatives de T, 

T 



r = 



V(n cos -h T sin 0) (ft* cos ^ + T* sin 0) 



1 f y y* I 

"*t"cos^ + T8in^"*"ft*cos^ + T*8in^j' 
et it Tautre limite, 



r = 



V(ft cos e + T sin 0) (ft* cos ^ + T*sin ^) * 

Or, en repi^sentant ces deux ^nations par 

?• = 22 — <^, r = 22, 
on trouve, pour la valeur de (log r) entre les deux limites, 

logi2-log(i2-^) = -log(l-|) = 0, 

i cause de la valeur infinie de 12. Ainsi la somme cherch6e est nuUe ; et il est tout 

clair que les sommes suivantes S / ^ » &c., se r^uisent de m6me k z^ro. 

(m, n/ 



Done enfin, 



20—2 



156 DEUXifeME MilMOIRE SUR LES FONCTIONS DOUBLEMENT PilRIODIQUES. [130 

Cela fait voir que 



u' = e"** M, 



le coefficient k ^tant donn^ au moyen de T^uation 

o{i la somme est prise, comme auparavant, entre les limites 

mod {(m, n) + y} > T, mod. (m, n) < T, r=oo. 

Mais il n'est pas permis d'^rire 

J J (m, n) 

En eflTet, cette intdgrale n'est que le premier terme d'une suite dont il faudrait, pour 
obtenir un r^ultat exact, prendre deux termes; le second terme de la suite serait 
une int^grale prise le long d'un contour, et il serait, ce me semble, tres-difficile d en 
trouver la valeur. Pour trouver la valeur de A, je remarque que k sera fonction 

lin^aire des quantity T^ y, y*, 4f , &c., qui entrent dans les valeurs de r ; done, 

puisqu'en demifere analyse I'=oo, k ne pent 6tre que de la forme Ly -{- My*, Cela 
^tant, en substituant pour u' sa valeur, je forme T^uation 

y(^+y) _ g_j«,t g(-.si/+Ly+iry)« n II + - 1 

y(y) * I {rn,n) + yy 

mod. {(m, n) + y]<T, r= oo , 
et j'&ris successivement 

ce qui donne pour les valeurs correspondantes du produit infini double e"**** . gx et 
€~****. Ox ; en comparant les valeurs ainsi obtenues avec les ^nations qui donnent les 
valeurs de y(a? + in), y(af + iT), on trouve 

2i = 0, M^ '^ 



mod. («i/ — oi't;) * 
ou enfin, 

y(a: + y) ^ ^_^^ /-^^mod. (l->-v)^)^ n(l+— ^— I 

y(y) ' 1 Kw) + yJ* 

mod. {(m, n) + y} < T, r= oo , 

laquelle est I'^uation qu'il s'agissait d'^tablir. II est & peine n^essaire de faire la 

X 

remarque que pour y = 0, on doit consid^rer & part le fiawteur 1 + - , lequel multipli^ 

if 

par y(y) devient tout simplement x\ T^uation subsiste done dans ce cas. 



130] DEUXIjkaiE Ml^OIBE 8UR LES FONCTIONS DOUBLEMENT PilRIODIQUES. 157 

En revenant au probl^me des transformations lindaires, partant des ^nations 

(2*; + l)ft, = Xfl+/AT, 
(2A: + 1)T, = i/n+/)T, 

je suppose d'abord que les coefficients X, v ne satisfassent pas k la fois aux deux 
conditions 

X = 0, 1/ = 0, mod. (2*; + 1), 
etje prends p, q des entiers quelconques tels, que \p-\-pq ne soit pas =0, mod. (2A; + 1). 
Cela ^tant, soient 

\p + vq =;>„ 
fMp + pq =gr„ 
(2*: + l)t=i>/"+3.T, 



et, par cons^uent, 



Je forme T^uation 



savour 



c'estri-dire 



\m + im — «p, = (2k -f 1) m^, 
fjLm + i/n — sq, = (2A; + 1) n^ , 



ou, ce qui est la m^me chose, 



n ^ sq = m^ fjL -- n^X. 

Or, m^, n^, ^ ^tant des entiers donn^, m, n seront aussi des entiers; de m6me, m, n 
^tant des entiers donnas, on trouve de A: i — i un entier a qui donne m^ un entier. 
V&is cela ^tant, n, sera aussi un entier; car autrement n^ serait une fraction ayant 
pour d^nominateur, lequel on voudrait, des nombres 2& + 1, X, i/, ce qui est impossible 
i moins que 

X = 0, V = 0, mod (2A; + 1). 

^^ si ces ^nations avaient lieu, on trouverait d'abord 8 de mani^re h, avoir n, entier, 
^t alors, puisqu'on n'a pas aussi 

/A = 0, p = 0, mod. (2* + 1) 

(^u effet, cela est impossible k cause de T^uation Xp — /ty = 2i + 1), on d^montrerait, 
«>nmie auparavant, pour n^, que m, est entier. Done, enfin, rw, n ^tant des entiers 
doim&, on trouve pour m,, n,, « un systfeme d'entiers tel que 8 soit compris de A: it — A, 
«t Von voit sans peine qu'il n'y a qu un seul systfeme de cette espfece. 



158 DEUXltME MISmOIBE SUB LES F0NCTI0N8 DOUBLEMENT pArIODIQUBS. [130 
A pr&ent, partant de T^uation 

y (y) I K. w,) + yj 

(et faisant attention k la particularity que prints le cas de y = 0), j'^ris successivement 

y-0. y = ±-<^,..., y = ±h^, 

et je forme le produit des ^nations ainsi trouv^e& Cela donne, k cause de (tn,, n,) + «^ 
= (m, n),, 

y(»f) I (»». «)/J 

la condition, par rapport aux limites, ^tant 

mod.(m, n),<T, T=ao. 



Or 






avec la m^me condition, par rapport smx limites; done, enfin, 



y^,.,-*t.,-^.«^..n|y^>|, 



oil, dans le num^rateur, s doit avoir toutes les valours enti^res depuis s^ — k jusqu'i 
« = + A:, y compris « = 0, et dans le d^nominateur ces m6mes valeurs, hormis la valeur 

8 = 0. 

II est, k present, facile de faire voir que cette propri^t^ subsiste pour I'une 
quelconque des fonctions yx, gx, Ox, Zx\ en eflTet, pour la d^montrer pour gx, jecris 
X + ^11 au lieu de a? ; en prenant, pour un moment, p = 2p' + 1, /a = 2/a', cela donne 



c*est-i-dire 



y, (*fi) 

Or, on d^uit de I'expression pour y^x, 

■ y,(ifi) 



y>+Jii) = eW.«(P.M),g^a,. 



= e-tf ,» (P. M) , w (B,-5+u) (*»+ite) n y (^ + *^ + i") 

= e-V.<* (p. M), £-4 (»,-5+i«)ir« ji dis+i) (Mte ij g (^ + ^Y*) . 



130] DBUXltME ICl^OIRE SUB LES FONCTIONS DOUBLEMENT P^RIODIQUES. 159 

ou enfin, k cause de T^uation 



la valeur de g^x est 

a a; = €-* (*.-^+^^«« . n g (^ + y . 
»' g («V^) 

et en repi^sentant, comme auparavant, Tune queleonque des fonctions yx, gx, Gx, Zx 
par Jx, on a T^uation 

equation dans laquelle 8 doit avoir, dans le num^rateur, t-outes les valeurs entiferes 
depuis « = — i jusqu'i 8 = k, y compris « = 0, et dans le d^nominateur, ees mSmes valeurs, 
hormis la valeur « = 0. 

Je suppose que les valeurs de p„ q, soient donn^es (cela va sans dire que Ton 
ne doit pas avoir k la fois p^sO, j, = 0, mod.2A: + l), et je remarque que Ton a, pour 
determiner X, fi, v, p, les conditions 

pp, - vq^ = 0, mod. {2k + 1), 

X=l, /A=0, mod. 2, 
v=0, p=l, 

Xp — fiv = 2k-\- 1. 

^^ cela ^tant, on aura ensuite, en rassemblant toutes les Equations qui ont rapport k 
la transformation, 

Pi>/ -»'?/ = (2* +!);>, 

(2A + l)n, = Xft +mT, 
(2A + l)T,= i/ft +pT. 



160 DEUXltME liiafOIRE SUR LES FONCTIONS DOUBLEMENT P^UUODIQUES. [130 

Or, quoique les valeurs de X, /i, v, p ne soient pas compl^tement d^termin^es au 
moyen de ces conditions, cependant il est clair que la valeur de la fonction J^x ne 
depend que des valeurs de p„ q, (en eflfet, ces valeurs suffisent pour determiner la 
quantity '4^=p^fl + gr^T, de laquelle depend la fonction J^x). Les formes diflKrentes de 
J,x, pour les systfemes de valeurs de X, ft, v, />, qui correspondent k des valeurs 
donn^es de p,, q,, doivent done se d^river de Tune quelconque de ces formes, au moyen 
dune transformation triviale des modules 11^, T^. II est, de plus, clair que les valeurs 
de p,, q,, qui sont ^gales k des multiples de (2A; + 1) pr^, ne donnent qu'une seule 
valeur de J,x. Je suppose d'abord que 

p, = 0, mod (2* + 1), 

on pent trouver un entier tel que 

0p,^l, mod.(2A: + l); 

en prenant alors 

0q, = q,, mod(2A? + l), 

cela donne 

(p,a + q,T,) = ft + qX mod. (2* +1), 
savoir 

^ = ft + qX Mttod. (2*; + 1). 

Mais en donnant k 8 des valeurs entiferes quelconques, depuis — k jusqu'& k, le syst^me 
des valeurs de syjt est equivalent au syst^me des valeurs de 80^, mod.(2A;+l); il est 
done permis d'^crire, sans perte de g^n^ralite, 

^ = ft + g,T. 
De m^me pour 

p, = 0, mod. (2k + 1), 

on d^montre que Ton pent donner k q, une valeur quelconque, sans changer pour 
cela la valeur de J^x\ il convient d'avoir p^ impair et q^ pair. J'^cris done, pour le 
premier cas, 2q^ au lieu de g^, et je suppose que, dans le deuxi^me cas, les valeurs 
de p,, q^ soient 

p, = 2A: + l, 5, = 2. 

Cela donne: 

Premier ca8. 

^ = ft + 2q,T, 

q^ un entier quelconque, y compris z^ro, depuis —A? jusqu'^ +A:. 

DeuxUme ca8. 
^ = (2ifc + l)ft + 2T; 

le nombre des valeurs diff^rentes de "9 sera done, en tout, 2A; + 2. 



130] DKUXliOIB M^OIBE SUB LES FONCTIONS DOUBLEMENT P^BIODIQITES. 161 



On obtient tout de suite, pour le premier cas, le syst^me d'^uations 

X = l, /*= 2q„ 

p = l, 5 = 0; 
^ = 2A+-l(" + 2q,T). 

Le cas particulier le plus simple est eelui de 9^ = 0; cela donne 



^ = "' = 2lTl"' '^' = '^' 



et, de 1^, 



11 



1 n 



T, 2ifc + l T' 

et m^me le cas g^ndral se r^uit k celui-ci, car, au moyen d'une transformation 
irimale, on obtiendrait 

ft' = n + 2q,T, T' = T. 

et puis 



Vr = ft,= 



2k + l 



ft', T, = r, 



et, de ]&, 



1 ft' 



ft 



T, 2k + lT' 



Les ^nations correspondantes pour le deuxi^me cas sont: 

p, = 2k+l, 9, = 2, 
\=2A+1, /t=0, 
V =0, p = 1, 

p=l. 9=2, 

^ = 2*1:1 t^^* "^ ^^ " "^ ^^ 
ft, = ft. 



T = 



' 2k + l 



T; 



^ qui donne 






c. n. 



21 



162 DEUXltME MJ^OIBE SUB LES FONCTIONS DOUBLEMENT PI^ODIQUES. [130 

J'ajoute, sans m'arrSter pour les d^montrer, quelques formules de transformation 
pour le nombre 2; je trouve d'abord 

Ces ^nations donnent, en introduisant les fonctions elliptiques, <f)x, fx, Fx donn^ au 
moyen de 

, yx - gx „ Gx 

^=fc' >=k' ^^=:^' 

les ^uations 

F,x ^ Fx ' 

dont la seconde peut encore s'^rire sous la forme 

et les deux ^nations combin^es ensemble conduisent sans peine & la valeur des 
modules c^, 6^. On trouve en effet, en mettant comme & Tordinaire 6' = c' + 6*, 

c/ = 46c, 

6/ = (6-c)», 
et puis 



- 1 — c (c — 6) i?x 
*'^= Wa; ' 

- _ 1 - c (c -h 6) ^'a ? 
•^'^■■l-c(c-6)<^«a?' 

F.x=^ 



l-c(c-6)^fl?' 

formules qui correspondent & celles de la transformation de Lagrange. Les ^nations 
pour y^a?, Z^x donnent encore une valeur de ^^a?, laquelle, ^gal^ k la valeur qui vient 
d'etre trouvfe, donne 

yx%xZ'^(\iX) _ if>xfx 

Z{x''\il)Z{x + Jft) " 1 - c(c-6) <^«a? * 



130] DEUXliaCE M^OIRE SUB L£S FONCTIONS DOUBLEMENT P^BIODIQUES. 163 

On obtient tout de suite les formules pour la transformation analogue il, = il, T^ = ^T. 
Mais il &ut de plus consid^rer le syst^me 



on aura alors 



et puis, en Anrivant 



on obtient 



ll, = i(n-T), T, = i(n + T): 



r yx = €~* <*'-*^^ gX Zx, 



-^_,-...-«.^y (^+in-T)y(5-4n-T) 
^' -yan-t) 



z* ( jn - T) 






-y»(in + T) 



^» (in + T) 

c,» = (c - tc)«, - e; = (e+ icy, 

Ax 
^'* ^fiTF'x ' 

, 1 +ice4^x 

J'" ~ fxFx ' 

l-tce4fx 



l + tce<ya; Z(a; + in-T)Z(a;-ifl-T) 
fx Fx Z* (iir^) gx Ox 



l-ice^fx Z(a! + ifl + T)Z(a;-^ + T) 
fxFx Z*{^ + t)gxOx 



\-\-iee^x ^ Z{x + \€l-T)Z{x-^-T)Z>(^ + r) ^ 
\-ice4fx Z {x + ^oTT) Z (x - ^T^t) Z* {^ -T)' 

01, an moins, ces formules seront exactes au signe de i prfes; car il serait peut-6tre 
*"ffi<ale de determiner quel est le signe qu'on doit domier h, cette quantity. 



21—2 



164 



[131 



131. 



NOUVELLES RECHERCHES SUR LES COVARIANTS. 

[From the Journal filr die reine und angewandte McUhematik (Crelle), torn. XLvn. (1854), 

pp. 109—125.] 

Je me sers de la notation 

(oo, ai,...an)(a?, yT 
' pour repr^enter la fonction 

en supposant que lea coefficients ao\ a^ &c. soient donnds par T^quation 

(oo, Oi, . . . On) (X^ + AAy, \'x + fiyY = (oo', Oi', . . . O (a?, y)\ 

suppose identique par rapport k x, y, soit ^(oo, ai,...an\ x, y) une fonction des co- 
efficients et des variables, telle que 

^K', Oi', ...On'; X, y) = (X/-X»P^(ao, ai,...an; \x + fiy, \'x'\-fify)] 

cette fonction ^ sera gdn^ralement un Covariantt et dans le cas particulier oil if> est 
fonction dea seuls coefficients, un Invariant de la fonction donn^. 

Je suppose d'abord que les nouveaux coefficients soient donn^ par T^uation 

. (Oo, Oi, ... an)(x + \y, y)~ = (ao', a,', ... an')(x, y)«; 
cela donne les relations 

Oi' = Gti + XOo, 

Oa' = ttj + 2\ai + X'Oo, 
&c. 



131] NOUVELLES BEGHERCHES SUB LES COVABIANTS. 165 

n £Biut done que le covaria/nt ^ satisfasse & T^uation 

4>W> <^> ... On'; a?, y) = <^(ao, Oi, ... a„ ; a? + Xy, y), 
laquelle peut aussi Stre ^rite comme suit: 

4>W> ai',...a„'; a?-Xy, y) = 0(ao, Oi, ...On; a:, y). (Z) 

De mSme, en fiEusant 

(Oo, Oi, ...a„)(a?, fix + yy^ioo", (h\...an")(x, y)«, 
ce qui donne 



On' =an 



le cooaruint ^ doit satisfaire aussi & I'^uation 

<t>(ao\ <h\'"(hi'\ a?, -/iic + y) = <^(ao, Oi, ...a„; «, y); (F) 

et r^proquement, toute fonction ^ homog^ne par rapport aux coeffieients et aussi par 
rapport aux variables, qui satisfait k ces ^nations (X, Y), sera un covariant de la 
fonction donn^. 

Examinons d'abord T^uation (X) que je repr&ente par ^' = 0. Soit pour le 
moment, Oi'— ai = Xai, a,' — a, = \aa, &c., alors on aura, comme k Tordinaire, T^quation 
symbolique 

oh les quantity Oi, Os, &c., en tant qu'elles entrent dans cti, a,, &c., ne doivent pas 
6tre affect^ par les symboles da^, 3a,, &c. de la differentiation. En substituant les 
valours de a^, a,, ...,et en ordonnant selon les puissances de \, cette ^nation donne 

ah les symboles D, Di, &c. sont donn^ par 

Q = ao3«, + 2aiaa, . . . + naj^ida^, 

et les quantity Oi, a,, &c., en tant qu'elles entrent dans les symboles Q, Di> &c, ne 
doivent pas Stre affect^ par les symboles da,, do,, &c. de la differentiation. II est 
assez remarquable que T^quation symbolique peut aussi 6tre ^rite sous la forme plus 
simple 

0' = e^D-y^.) <^, 



166 NOUVELLES RECHERCHES 8UR LE8 CO VARIANTS. [131 

oh lea quantit^s Oi, a,, ..., en tant qu'elles entrent dans le symbole D, sont cens^es 
affect^s des symboles do,, d^,, &c. de la diff(^rentiation ; de mani^re que dans le d^veloppe- 
ment, D'.0 par exemple, signifie D . D0, et ainsi de suite. Je ne m'arrSte pas sur 
ce point, parce que pour ce que je vais d^montrer de plus important, il suffit de faire 
attention a la premiire puissance de X. D'ailleurs Tintelligibilit^ des ^nations dont 
il s'agit, sera facilitde en faisant les d^veloppements et en comparant les puissances 
correspondantes de \. Cela donne par exemple: 

n«=g«+2n„ n»=n»+3nni+6n„ &c. 

oil les symboles D^ D^ &c. k gauche de ces Equations d^notent la double, triple, &c. 
rdpdtition de reparation D, tandis qxx'k c6t^ droit des Equations, les quantity Oi, a,,... &c., 
en tant qu'elles entrent dans les symboles D, Di, &c. sOnt cens^ ne pas Stre 
affect^es des symboles da^, do,, &c. de la diff(^rentiation. Dans la suite, si le contraire 
n'est pas dit, je me servirai des expressions D', D", &c. pour d^noter les repetitions de 
rop^ration, et de m6me pour les combinaisons de deux ou de plusieurs symboles. 

Cela etant, T^quation 0' = c^(D-i'»J ^ = donne 

<^={i+x(n-ya,) + j^*2^n-ya,)»+...}0, 

oil (□ — y3«)'.0 (JQ le r^pfete) equivaut a (D ""y3«)'(g ~y^«)^> ®^ *"^ d® suite. H 
faut d'abord que le coeflScient de X s'^vanouisse, ce qui donne (D— y3jB)^ = 0; et cette 
condition ^tant satisfaite, les coefficients des puissances sup^rieures s'^vanomssent d'elles- 
mSmes ; c'est-&-dire, T^quation (X) sera satisfaite en supposant que ^ satisfait k I'^quation 
aux differences partielles (D — ydx) <f> = 0, 

EIn posant « 

□ = ctn3«^_j + 2a^i3a^ . . . + noida^, 

on fera un raisonnement analogue par rapport k Tdquation (F); et il sera ainsi demontre 
que <f> doit satisfaire aussi k r^quation k differences partielles (D — «9y)^=0; done 
enfin, on a le suivant 

TH]£oBi3iE. Tout covariant ^ de la fonction 

(ao, Oi, ... an){x, y)~, 
satisfait aux deux Equations k differences partielles 

(n-ya,)0=o, (n-^y)<^=o, (.i) 

oil 

« 



131] NOUVELLES RECHERCHES 8UR LES OOVARIANTS. 167 

et r^proquement toute fonction, homog^ne par rapport aux coefficients et par rapport 
aox variables, qui satisfait k ces ^uations, est un covariant de la fonction donn^. 

Par exemple, Vinvariant ^^clc — I^ de la fonction cux^ + 2hxy + cy* satisfait aux 
Equations 

et le catfariant = (ac — 6*)«'+(a9 — 6c)a?y + (63 — c*)y* de la fonction aa:^+Sba^y-\-3cxi^+dy* 
satis&it aux Equations 

(aa^ + 2Me + 3cad-ya,)0 = O, (3d0e + 2ca6+W«-ajay)<^ = O. 

n est clair qu'en ne consid^rant que les fonctions qui restent les mSmes en prenant 
dans un ordre inverse les coefficients Oq, Oi, ... On et les variables x, y, respectivement, 
les covariants seront d^finis par Tune ou I'autre des Equations (A), et qu'il n'est plus 
n^cessaire de consid^rer les deux Equations. Cela posd, on trouve assez facilement les 
comariants par la m^thode des coefficients inddtermin^. Mais il y a & remarquer une 
circonstance de la plus grande importance dans cette throne, savoir, que Ton obtient 
de cette mani^ un nombre d'^uations plus grand qu'il n'en faut pour determiner 
les coefficients dont il s'agit Ces Equations cependant, ^tant li^es entre elles, se r^uisent 
au nombre n^cessaire d'^uations ind^pendantes. 

Cherchons par exemple pour la fonction a^ + 3ha^y + 3cd7y' + dy* un invariant ^ de 
la forme 

= ila«* + 5a6cd + Ooc' + Gh'd + D6«c», 

contenant les quatre coefficients ind^termin^s A, B, C, D, £n substituant dans T^quation 
(085+ 263c + 3cdd) ^ = 0, on obtient 

(3C + 25) a6»d + (3B + 6(7 + 2Z))a6c> + (6^ + 5) ac»d + (3(7 + 42)) 6»c = ; 

Of les quatre ^nations donn^es par cette condition, se rdduisent k trois Equations 
ind^pendantes, de sorte qu'en fsdsant par exemple -4=— 1, les autres coefficients seront 
d^tennmds, et Ton obtient le r&ultat connu : 

(^ = - d^« + 6a6cd - 4ac» - 46»d + Wc\ 

La circonstance mentionn^ ci-dessus s'oppose k r&oudre de la mani^re dont il 
^^t, le probl^me de trouver le nombre des invariants d'un ordre donn^: probl^me 
^ a toujours brav^ mes efforts. 

Avant d'entamer la solution des Equations {A)y je vais d^montrer quelques propri^t^s 
^^rales des covariants, et des invariants. Pour abr^ger, je me servirai du mot pesanteur, 
en disant que les coefficients Oo, Oi, &c., ont respectivement les pesanteurs — ^n, 1 — ^n, 
^) que les variables x, y ont respectivement les pesanteurs ^, — ^, et que la pesanteur 



168 NOUVELLES RECHEECHES 8UR LES OOVARIANTa [l3l 

d'un produit est ^gale k la somme des pesanteurs des facteuis. Cela pos^, je dis que 
tout covariant est compost de termes dont chacun k la pesanteur z&o. Pour d^montrer 
cela^ j'^cris: 

(n-a0y)(n-ya,) = nn-ya,D-aj0yn+«ya.ayH-«a,; 

cela donne 

« 
or, en faisant attention aux valours de D, D, savoir 

gn = (DD) + nooaa^ + 2(n - l)a,a«^ ... + n 1 a^,a«_^, 

oil, en formant les produits (DU), (UD)) I^ quantit^s Oo, Oi, ... o^ sont cens^ non 
affect^s par les symboles da^,da^,... da^ de la difii^rentiation, on en tire 

nn-nn = nao3a +(n-2)oi9a ... — nonda 

= - 2 {(0 - hn)aoda^ + (1 - ^n)a,da^ . . . + (n - in)anda,} = - 26, 
en reprdsentant par 6 Texpression symbolique entre les crochets. De \k enfin on obtient : 

Or en supposant les deux parties de cette ^nation symbolique appliqu^ au covariant 

^, la partie gauche de T^uation s'^vanouit en vertu des ^nations (A) et T^uation 

se r^uit k 

(8 + ia«x-iyay)0 = O; (B) 

ce qui est une nouvelle Equation k differences partielles, k laquelle satisSut le covariant 
<!>. II est ais^ de voir que cette Equation exprime le th^r^me dnoncd ci-dessus, 
savoir que tout covariant est compost de termes de la pesanteur ziro. 

n suit de Ik, en considdrant un covariant 

<^ = (ilo, Au ... At)(x, yy 

qu'un coefficient quelconque At aura la pesanteur i — ^8, ou bien que les pesanteurs 
ferment une progression arithm^tique aux diff(^rences 1, et dont les termes extremes 
sont — i«, +^s. 

Substituons maintenant cette valeur de <f) dans les ^nations (A). La premie 
^nation donne d'abord: 

nilo = 0, DAi^Aoy DA^^iAu ...nA, = sA^^ (a) 

Cela est un syst^me qui ^quivaut aux deux Equations 

n'.il, = 0, = y'.e°f.il, (cT) 



131] NOUVELLES RECHERCHES 8UR LES CO VARIANTS. 169 

De mSme, la seconde ^uation donne 



BTSt&me qui ^uivaut aux deux Equations 

[!l'+Mo = 0, 4>^afe^».Ao (/SO 

On voit que A^ satisfait aux deux ^uations 

nA = 0, 6'+^ilo = 0, (7) 

et en supposant que cette quantity soit connue, on trouve les autres coefficients 
^1, A^ ...,Ag par la seule differentiation, au moyen des ^uations 0^). Or cela dtant, 
je dis que les ^nations (a) seront satisfiedtes d'elles-mSmes. "En. effet: des Equations 
Dilo=0, [!lulo=«ili on tire 00^=0, DDA^^sDAu et de ]k (DD - nn)ilo = -«nili. 

Or nous avons d4jk vu que CD — 00 = 26, et TAjuation (B) donne B.Aq+^s.Ao^Oi 
done r^uation (DO — OiJ)ilo = — «0-4.i se r^uit k Ao=QAi: ^nation du systfeme 
(a). De la mSme mani^re on obtient les autres ^nations de ce syst^me. On pent 
dire que Ton aurait pu determiner ^galement le coefficient Ag au moyen des ^nations 



D4. = 0, D'.As = 0, (8) 

et de ]k les coefficients il^i, ... Aq par les ^nations (a). 

Prenons par exemple un covariant (Ao, Ai, A^) (a?, y)' de la fonction cubique 
(w* + 36a^ + 3ca?y* + dy*. A^ doit satisfaire aux deux ^nations 

(adt + 2hdc + ^d)Ao = 0, (36a« + 2cdt, + ddcfAo = 0. 
Ces Equations sont en effet satisfiedtes en mettant Ao = ac — b^. On a done les Equations 

2Ai = {3bda + 2cdb + ddc)Ao, A^ = (Sbda + 2ddt + ddc)Au 

pour determiner Ai, A^; ce qui donne 2-4i = ad — 6c, il, = 6d— c", et on est conduit 
unsi au covariant mentionn^ ci-dessus, savoir k 

(ac - b^)a^ + (ad -be) ay + (bd - c^) y\ 

Soit maintenant 

ai^da^-'Of^^yda^ ... ±r3a.=A, 

on aura 

nA = (gA), AD=(AD)-y||. 
C. n. 22 



170 NODTVELLES RBCHERCHES SUR LES COVARIANTS. [131 

oil dans (DA), (AD) les quantity Oq, Oi, ... sont cens^ non affectdes par lea symboles 
da,> do,, &C. de la di£fdrentiatioD. Cela donne 

DA-AD = y^. 

Or 3aj A — Adx = ^ , done : 

(D-ya.)A = A(D-ya,), 
et de m6me: 

(n-aj0y)A = A(n-a»yX 

Appliquons ces deux Equations symboliques k un covariant 0. Les termes k droite 
s'^vanouissent k cause des ^nations (A), et Ton obtient les deux ^nations 

. (D-ya«)A0=O, (n-'/cdy)A4>^0, 

o'est-ji^dire : A0 sera aussi un covariant de la fonction donn^e. Par exemple de Vinva- 
riant 

on tire le covariant 

savoir : 

(- a^ + 3a6c - 2i» )«» 

-3( aM-2ac«+ 6«c)aj^ 

+ 3( acd-26«dH- 6c«)ay« 

- (-a(?+3acd-2c» )y»; 

r&ultat ddjd. connu. 

Essay ons maintenant k int^grer les ^nations {A)\ savoir: 

(n-y9«)0-O, (D-aj0y)^ = O. 

Pour int^grer la premiere, je reviens k une notation dont je me suis ddjij servi dans 
ce mdmoire et j'^cris 

Oj' ssOi + XOo, 

Oa' = a, + 2\ai + X'Oo, 

On' = On + w\a»-i • • • + X**ao. 



131] NOUVELLES RECHERCHES 8UR LES OOVARIANTS. 171 

En faisant X = , ce qui donne a^ = 0, on voit sans peine que Ton satisfera ^ 

r^uation, en mettant pour <^ une fonction quelconque de quantitds Oq^ a^, ... a^^ 
^ + ^y> y \ 1® nombre de ces quantity ^tant n + 2. Et cela est la solution gdndrale de 
r^uation. 

Ce r&ultat doit 6tre substitute dans la seconde Equation, savoir dans (U — ^y)<^=0. 
Four cela, imaginons que les quantitds Oo, Oi, ... On, ^, y soient exprimdes en fonction 
de Oo'f CLtt ..• fltn'i ^> y et Ci ; puisque ^ est fonction des seules quantitds Oq', a^ ... On', 
X, y, r^uation r^sultante doit 6tre satisfiedte, quelle que soit la valeur de Oi. Or on 
trouve que cette Equation r&ultante a la forme L + Mai=:0: done il faut qu'on ait 
k la fois les deux ^nations Z = 0, M=0. (Je renvoie k une note les details de la 
r^uction.) En demi^re analyse, et en remettant dans les ^nations X — 0, M = les 
quantit^s Oo, a,, ..., On au lieu de Oo^ o^^ ..., an\ je trouve les r^sultats suivants tr^ 
simples, savoir, en ^crivant 

e = (0 - ^n)a,da^ + (2 - ^n)a,d^^ + (3 - in)a,a«^ . . . + (n - in)anda^ : 



•D = (n - 2)a^da + (n - S)a,da ... +and, 



«-i" 



Les ^nations dont il s'agit sont 

{(n-l)a,en-ya,)-aoCn-^,)}^=0, (C) 

(8 + ia«,-Jyay)^=0, (D) 

et il y a & remarquer qu'on obtient T^quation (C) en ^liminant entre les ^nations 
(A) le terme doi^; ^t puis, en mettant ai = 0, on tire I'^uation (D) de T^uation 
(B), en y mettant de mSme Oi = 0. II y a & remarquer aussi que la fonction <f> qui 
Kitis&it aux ^nations (C, D), est ce que devient un covariant quelconque <^, en y 
ntettant Oi^O. On obtient d'abord la valeur g^n^rale en changeant Oo, a,, ... , On en 
^'> 0,', ..., On^ et en mettant apr^ pour ces quantit^s leurs valeurs en termes de 
^f Oi, 0,, ..., On- La solution du probl^me des covariants serait done effectude si Ton 
pourrait int^grer les Ajuations (C, D). 

Or la quantity Oq entre dans Tdquation ((7) comme constante, et Ton voit sans 
peine que cette ^nation pourra 6tre int^grde en mettant Oo = 1 ; puis, en ^rivant dans 

le r&ultat — , — , ... — au lieu de Oj, 0^, .., a^ et en multipliant par une puissance 

qnelconque de Oq, le r&ultat ainsi obtenu, serait composd de termes de la mdme 
t^nteur; et en choisissant convenablement la puissance de Oo, on pourrait faire en 
sorte que ces termes fussent de la pesanteur z4to. Mais Tdquation (D) ne fait qu'exprimer 

que la fonction ^ est compos^e de termes de la peaanteur z^ro; le rdsultat obtenu de 
^ mani^re dont il s'agit, satisfera done par lui-m6me k Tdquation (D), et il est 
pennis de ne fiedre attention qu'& I'^quation (C). Dans la pratique on intdgrera cette 

22—2 



172 N0UVELLE8 BECHEBCHES 8UR LES GOV ABI ANTS. [131 

^nation en ayant soin de faire en sorte que les solutions soient de la peswnJteur z^ro, 
ce qui peut Stre effectu^ en multipliant par une puissance convenablement choisie de 
Oo. Et puisqu'en faisant abstraction de cette quantity Oo, T^quation {(J) contient n + 1 

quantit^s variables, savoir a,, a,, ...,an, x, y, la fonction ^ sera une fonction arbitraire 
de n quantity ; et en supposant que cette fonction ne contienne pas les variables 
w, y (cas auquel serait ce que deviendrait un invariant quelconque en y mettant Oi = 0), 

^ sera une fonction arbitraire de n — 2 quantity. 

La mSme chose sera ^videmment vrai, si Ton r^tablit la valeur gdn^rale de a^ : 
done tout invariant sera une fonction d'un nombre n — 2 dUnvariantSy que Ton pourra 
prendre pour primitifs; et tout covariant sera une fonction de ces invariamts primiti& 
de la fonction donn^ (laquelle est dvidemment un de ses propres covariants), et d'un 
autre covariant que Ton peut prendre pour primitif. Cela ne prouve nullement (ce qui 
est n^nmoins vrai pour les invariants, k ce que je crois) que tout invariant est une 
fonction rationnelle et int^grale de n — 2 invariants convenablement choisis, et que tout 
covariant est une fonction rationnelle et int^grale (ce qui en effet n*est pas vrai) de 
ces invariants, de la fonction donn^e, et d'un covariant convenablement choisL 

Le cas n = 2 fait dans cette th^rie une exception. On sait qu'il existe dans ce 
cas un invaria/nt, savoir ac^l^ qui, selon la th^rie g^n^rale, ne doit pas exister, et 
il n'existe pas de covariant, hormis la fonction donn^e elle-mSme. Or cette particularity 
peut 6tre aisdment expliqu^. 

Le cas 71 = 3 rentre, comme cela doit 6tre, dans la th^rie g^n^rale. En effet, il 
existe dans ce cas un invariant, savoir la fonction — a'cP + 6ahcd + 4ac* — 46'd + 36'c* 
ci-dessus trouv^, et tout covariant de la fonction peut 6tre exprim^ par cet invariant 
de la fonction donn^ elle-mdme, et par le covariant (oc — 6*)«' + (ad — 6c)a;y + (M — c*)y' 
ci-dessus trouvd II en est ainsi par exemple pour le covariant de troisi^me ordre 
par rapport aux variables et aux coefficients; car en repr^ntant par ^ le co- 
variant dont il s'agit, par H le covariant du second ordre, par u la fonction donn^ 
aa? + Zbah/ + 3ca;y' + dy* et par V Vinvariant, on obtient I'^uation identique 
^' + Dw* = — 4jEr". Je £aas mention de cette ^nation, parce que je crois qu'elle n'est 
pas g^n^ralement connue. 

Je vais donner maintenant quelques exemples des Equations (C et Z>). Soit d'abord 

71=3, et supposons que 4> ^^ contienne pas les variables x, y: ^ sera une fonction 
de a, c, d, et les Equations reviendront k 

{Qd'dd - adde)^ = 0, (- 303^ + 09^ + 3(©d)^ = 0. 

Les quantit^s oc*, a*d^, dont chacune est de la pesanteur ziro, satisfont par 1& d. la 
seconde Equation, et en mettant ^ = ila'cP + Coc', on obtient 4il — C=0, en vertu de 
la premiere Equation; ou en faisant il = — 1, cela donne C= — 4; de Id. on tire 
^ = — a*cP — 4ac*, et la solution gdn^rale est $ = jF(— a'cP — 4ac*), F ^tant une fonction 
quelconque. La formule plus g^n^rale ^^F{a, ^a^d^—^Kuf) satisferait sans doute k la 



131] NOUVELLES RECHERCHES SUR LES CX) VARIANTS. 173 

premie ^uation, mais pour que cette valeur satis&sse k la seconde ^uation, il faut 
que la quantity a, en tant qu'elle n'est pas 'contenue dans — a^c^ — isoc^, disparaisse. 
Ainsi la valeur donn^ ci-dessus, savoir ^ = jF(— a*(? — 4ac*), est la solution la plus 
gt^n^rale des deux ^nations. 

Ecrivons a, c , d 1 — r au lieu de a, c, d, et 6 au lieu de $, nous obtenons : 

4> = F{'-a?d} + 6abcd - 4ac» - 46W + 36V); 

ce qui est Texpression la plus g^n^rale des invariants de la fonction aic*+3&ij^+3ca:^*+y', 
et Ton Yoit que tous ces invariants sont fonctions d'une seule quantity que nous avons 
prise ci-dessus pour Vinvariant de la fonction de troisi^me ordre dont il s'agit 

Soit encore n = 4, sera une fonction de a, c, (2, e qui satisfait aux Equations 

{2acBc + (a« - 9c*) da - licdde] ^ = 0, 
{-2aaa + d3d + 2eae}^=0, 

dont la solution gdn^rale est = F{ae + 3c*, axie — acP — c*), F ^tant une fonction quel- 
conque. On voit par Ul qu'il n'existe que les invariants ind^pendants ew — 4cd + 3c*, 
ace + 2icd — ocP — 6*6 — c*. Ce r&ultat est connu depuis longtemps. 

Soit enfin n = 5, <^ sera une fonction de a, c, d, e, f qui satisfait aux Equations 

{3adde + (2ae - 12c*) da + (a/- 16cd) de - 20cedf] ^ = 0, 

On sait qu'il y en a une solution de quatri^me ordre par rapport aux quantity 
a, c, d, e, f\ et en prenant la fonction la plus g^n^rale dont les termes ont la pesanteur 
z^ro, on aura: 

= ula*/» + Bacdf+ Cac^ + Bad^e + Ec^e + F(M^ : 

fonction qui satis&it d'elle-mSme k la seconde Equation. En substituant cette valeur 
dans la premiere ^nation, on trouvera que les coefficients A, B, &c. doivent satisfaire 
i ces sept ^nations : 

2£ + 2C-40il = 0, 35 + 2) = 0, 3C + 42) = 0, -125 + iE' = 0, 
QiF- 24Z> + 4^*- 32C - 205 = 0, 6i^-162)=0, - 24i^- 16iE'= 0, 

<I^ se r^uisent cependcuit (ce que Ton n'aurait pas facilement devin^ par la forme 
des ^juations) k cinq Equations ind^pendantes. En faisant done il = 1, on trouve 
aia^meiit les autres coefficients B, G, &c. et on obtient ainsi : 

^ = a*/» + 4acd/+ 16acc* - 12ad*e + 48c»c - 32c*d* : 

▼aleur qui pent 6tre tirde d'une formule prdsent^e dans mon m^moire sur les h3rper- 
d^rminants, [161 ^^ y f^^dsant 6 = 0. 



174 NOUVELLES BECHERCHES 8UR LES COVARIANTfiL [l31 

J'ai donn^ cet exemple pour faire voir qu'il serait impossible de deduire du Dombre 
suppose connu des coefficients inddtermin^s qui correspondent, k un ordre donnd, le 
nombre des invariants de ce mSme ordre. II est done inutile de pousser plus loin cette 
discussion. 



Note 1 8ur Vintigration des iqwiUons {A). 

En ^rivant comme ci-dessus: 

n = afia, + Soiaao. ... + nan-i3a,, 

n = noxda. + (n - \)a^da, ... + CLn\_^^ 

il s'agit de trouver une quantity ^, fonction de Oq, Oi, ... an, a? et y qui satisfasse k 
la fois aux ^nations 

(n-a?ay)0=o. 

Pour int<^grer ces ^nations, j'^ris, comme plus haut : 

do' = 00, 

a/ = ai + Xoo, 
ai^^a^-^ 2Xai + Va©, 



On' = On + nXa,i_i ... +X*ao, 

et aussi of = x — \y, y' = y. Cela pos^, je fais remarquer d'abord que ^r- = ao'» ^=2ai', 

et ainsi de suite. Eln consid^rant X comme fonction quelconque de Oo, Oi, ... On, et en 
supposant que ^ soit une fonction de Oo', Oi^ ... a^, oiy y\ on parvient assez facilement 

k r^quation identique (D — y9aj)^ = (l + D^) (III'~y'3«')0> ^^ Q' ©st ce que devient 
D, en y ^rivant Oo', Oi', ... a^ au lieu de Oq, Oi, ... On. 

Nous pouvons done satisfaire k la premiere ^nation, en determinant \ au moyen 

de l + nx=0: &}uation qui serait satisfaite en ^rivant \= — — , ou, si Ton veut, en 

determinant X par a^^O. Done, en supposant toujours que X ait cette valeur, 4> 
sera une fonction quelconque de Oo', a,', ... a^, of, y\ c'est-i-dire d'un nombre n + 2 
de quantit^s. Ce sera done \k (comme on aurait pu facilement prdvoir), la solution 
g^n^rale de la premifere Equation. Or en consid^rant <f> comme fonction de Oo', a,', ... a„\ 
of, yfy ou, si Ton veut, de Oo', a^, a,', ... a^, of, y' (oil Oi' = ai + Xo© = 0), et en 



131] NOUVELLES RECHERCHES SUB LES COVAMANTS. 175 

sabstituaDLt cette valeur dans T^quation (D — a^y)0 = 0, on voit d'abord que la variation 
de la quantity X fonmit au r^sultat le terme 



(„a.gH«-l)«.£)(n'-j^.)*; 



et puisque na ^ + (^""^)^^'^ r^uit & n-^-(n — 1) — ^, ou enfin & V-^^ —, 

ce terme devient 

Le terme —oi'dy,^ se r^duit & — (a?' + XyO(""^«'+3y')^» savoir & 

et en mettant pour un moment 

M— woi(9fl,' + X9aj' +X*3|,^/) 

+ (n-l)a2( ao,' +nX«-^aa;) 

+ an( 9a^/+wXa«^,), 

nous obtenons 
c'est-i-dire 

ۥ0 

Or en supposant que D' est ce que devient D en y ^crivant a©, Oi', ... a»' au 
lieu de Oo* ai, ... On, et en posant 

e' = (0 - in)ao'aa.' + (1 - in) Oi'ao,^ + . . . (n - iii)an'aa;, 

on obtient, aprte avoir fait une r^uction un pen pdnible : 

M^ + X«n'<^ = n'0 + 2X8'^, 

(en effet les coefficients de da,'<^, 'da{^ &c. aux deux cdtes de cette ^nation deviennent 
les mSmes aprte des r^uctions convenablea) Done enfin on a 

(D - a0v)* = ([i' - a;^,^)^ - ^^^^^^ (D' - y'3^)^ + 2X(e' + J<c^^ - iy'8^)^ = 0. 

ott bien, puisque cette Equation doit Stre satis&ite inddpendamment de la quantity X (qui 
fleule contient Oi), elle se decompose dans les deux ^nations 

{aii^ - x'Z^) - (n - 1) ai{U' - j/a^)} <^ = 0, 



176 NOUVELLES RECHERCHES SUB LES COVARIANTS. [131 

lesquelles, en y mettant d'abord a^^O, puis en remettant Oo, a,, ..., On, a?, y an lieu de 

Oo', cLtf ... , Oni «^» y'l et en ^crivant 0, O, •Q, d au lieu de ^, O, D, D, donuent en 
effet les Equations (C, D) dont je me suis servi dans le texte. 



Note 2. 

Je vais r^sumer dans cette note quelques formules qui feront voir la liaison qui 
existe entre les invariaiUs d'une fonction de fi-i^me oidre et de la fonction de (n — l)i&me 
ordre que Ton obtient en r^uisant & z^ro le coefficient de y*^, et en supprimant le 
facteur x. 

IV convient pour cela de consid^rer une fonction telle que 

(Oo, Oi, ... an){^,y)n = a^ + (h^~^y ... +a„y~, 
dans laquelle n'entrent plus les coefficients numdriques du bindme (1 + x)\ 

Ecrivons 

{a^,Q^, ... a„)(a?, y)« = ao(a?-aiy)(a?-a^) ... (a?-any); 

je t&che d'abord & repr&enter les invariants au moyen des racines ai, a„...,a„, et 
j'^tends pour le moment le terme invariant k toute fonction, sym^trique ou non, des 
racines qui ait la propri^t^ caract^ristique des invariants: fonctions qui jusqu'ici ont 
4t6 consid^r^es tacitement comme rationnelles par rapport aux coefficienta 

Mettons d'abord 

V = Oo^^ioi - a^fioi -«,)»... ((V-i - OnY ; 

cette quantity V qui, ^gal^ k ziro, exprime T^galit^ de deux racines, et que je vais 
d^rmais nommer le Discriminant de la fonction, sera une fonction rationnelle des 
coefficients, et d'un invariant proprement dit. Mais de plus, toute fonction telle que 
(Oi — a,)**(ai — a,)*, ..., dans laquelle la somme des indices des facteurs qui contiennent 
ttx, celle des indices des facteurs qui contiennent a,, &c. sent ^gales, sera un invariant; 
et en r^unissant ces fonctions, pour trouver une somme en fonction 8}anetrique des 
racines, on obtiendra des invariants proprement dits. Cela soit dit en passant. Pour 
le moment il suffit de prendre les invariants les plus simples, savoir ceux de la forme 

(«! - tt») (g| - q^) 
(ax - a,) (a, - fl^i) ' 

lesquels en effet sent des rapports anharmoniques de quatre racines, prises k volontd. Soient 
Oi» Os, ••.>0»-i 1* fonction qui vient d'etre ^crite et les fonctions que Ton en tire en 
mettant a^, fl^j, ...,an au lieu de a^. Les fonctions V, Q^, Qj, ...,Q,i_, seront des 
invariants ind^pendants, et le nombre de ces invariants est n — 2. Done, tout autre 



131] NOUVELLES RECHERCHES SUB LES CO VARIANTS. 177 

invariant sera une fonction des quantity V, Q^, Q,, ..., Qn^, Soit maintenant an = 0» 
et cEn 1ft racine qui devient dgale k z4ro. Lea qucuitit^ Qi, Qa* •••> Qn^ seront toujours 
des rapports anharmoniques de quatre racines de T^uation da (n — l)i6me ordre. II 
n'y aura que la seule quantity Qn^ qui change de forme, et elle ne sera pas un 
intfariant de la fonction du (n — l)i^me ordre. On voit aussi d'abord que le discriminant 
V se r^uit k a*n-iVo» en exprimant par V© le discriminant de la fonction du (n — l)ifeme 
ordre. (Cest je crois M. Joachimstbal qui a le premier remarqud cette circonstance.) 
Done, en supposant an = 0, Vinvariant de la fonction du n-i^me ordre deviendra une 
fonction de a^n-^o, Qi, Q,, ... Qn^ et d'une quantity X qui n'est pas un invariant de 
la fonction du (n — l)ifeme ordre, mais qui sera toujours la mSme quel que soit Tinvariant 
dont il s'agit. En consid^rant les invariants proprement dits de la fonction du (n — l)i^me 
ordre, on pent former avec ces irivariants des quotients /i, /„ ...,/,^-^ du degr^ z^ro 
par rapport aux coefficients. Nous pouvons remplacer par ces quotients les quantit^s 
Qii Qt) "-iQnr-ii et dire que Vinvariant de la fonction du n-i^me ordre, en mettant an = 0, 
deviendra une fonction des quantit^s a'»_iVo, A, /,, ..., /,^-^ et X. 

Ces thdor&mes auront, je crois, quelque utility pour les recherches ult^rieures: je 
les laisse k c6t^ maintenant, et veux presenter une m^thode assez simple pour calculer 
les discriminants. 

Four cela je remarque que les Equations {A\ en changeant, comme nous venous 
de le £Edre, les valours des coefficients, donnent pour les invariants : 

(naoda, + (n - l)aida, . . . + a^^ida) = 0, 

(aida. + 2a,'aa, • • • + nanda^)4> = ; 

et ces ^nations seront satisfiedtes en mettant pour <f> le discrimina/tU V. Or, pour 
fln=0, la fonction V devient a*«_iVo, ou, si Ton veut, — a'n-iV©; done V sera g^nerale- 
nient de la forme 

oil On*^^ est la puissance la plus dlev^ de an- Done, en supposant que Vq soit connu, 
et en mettant la premiere des Equations Writes ci-dessus sous la forme (^+a»_i3a^)V=0, 

0^ l'=na©9ai + (n — l)oi3fl, ... +2aw»,3a , on obtiendra par la seule diff(^rentiation les 

efficients B, C, &c. EIn effet, cette Equation donne 

a^iB^F{a*^,V,l %i^,G — F{B\ San^,D = ^ F(C) ; 

et ainsi de suite. 

En supposant par exemple n = 3, consid^rons la fonction du troisi^me ordre 
^^ ixfmminani de okc* + ^xy + yy* sera 4ay — /S*. Nous avons alors 

an. 23 



178 NOUVELLES BECHERGHES SUB LES CJOVABIANTS. [131 

et en mettant ^=3a9^ + 2i98y, B, C seront donn^ par 

c'est-Ji-dire B ^ ISafiy - *fi*, C = -27a», et de ]k: 

V = - 27 fl?S« + ISa/SyB - 407' - 4/9»S + ^Sy : 
valeur qui correspond en efifet k la forme ordinaire 

V = -a«* + 6a6od-4ac«-4W + 86V, 
en changeant d'une mani^re convenable les coefficient& 

Londres, SUme Buildings, 23 Fhr. 1852. 



132] 



179 



132. 



REPONSE A UNE QUESTION PROPOSEE PAR M. STEINER 

(Au%abe 4, Crelle t. xxxi. (1846) p. 90). 



[From the Jcumal far die reins und angewandte Mathematik (Crelle), torn. L. (1855), 

pp. 277—278.] 

En partant des deux th^rfemes : 

I. Qu'U existe au moins une surface du second ordre qui touche neuf plans donn& 
quelconques ; 

n. Que le lieu d'intersection de trois plans rectangles qui touchent une surfisM^e 
^^ second ordre est une sphere concentrique avec la surface, tandis que pour le para- 
^loide cette sphere se r^uit It un plan, 

^ Steiner suppose le cas d'un parall^epipkle rectangle, ou mdme d'un cube P 
^^ d'un point quelconque D, par lequel passent trois plans rectangles. Les six plans 
^^ parall^lepipMe P et les trois plans qui passent par le point D seront touchy d'une 
^^'fece F du second ordre (I.), et les huit angles E du parall^epipfede P et le point 
^ doivent done se trouver tons les neuf sur la surfiek^e d'une sphere, ou dans un 
P^^^ (XL). Les huit angles E sont en efifet situ6s sur la surfiek^e d'une sphere, 
^^t^rmin^ par eux ; mais le point D ^tant arbitraire, ce point en g^n^ral ne sera 
P^B situ^ sur cette surface sph^rique, de mani^re que les neuf points 8E et D ne 
^<t>nt ntuds, ni dans une surface sph^rique, ni dans un plan; ce qui ne s'accorde 
P^ avec le th^orfeme II. Cela etant, M. Steiner dit, qu'il y a It prouver que la 
^xitradiction n'est qu'apparente, et que tout cela n'affaiblit pas la validity g^n^rale des 
deux th^rfemea 

II s'agit de savoir ce que devient dans le cas suppose par M. Steiner la surface 
du second ordre qui touche les six plans du parall61epipMe P et les trois plans qui 

23—2 



180 lUiPONSE 1 UNB QUBSnON PROPOSJ^ PAR M. 8TEINER. [132 

passent par le point D. Cette surfSeu^ sera en eifet la conique aelon laquelle Vinfini, 
consicUrS comme plan, est coupi par un cdne ditermind, pris la position du sommeL 
En efifet, menons par un point queloonque de Tespace trois plans parall^les auz plans 
du parall^epipMe P, et par le point D trois autres plans parall^les k ces plana Ces 
six plans seront touch& (en vertu d'un th^rfeme oonnu) par un cdne ddtermin^ du 
second ordre, et on pent dire que ce cdne, quelle que soit la position de son sommet, 
rencontre Tinfini, consid^rd comme plan, dans une seule et m£me conique (cela n'est 
en effet autre chose que de dire que deux droites parallfeles rencontrent I'infini, con- 
siddr^ comme plan, dans un seul et m£me point). Le cdne dont U s'agit aura la 
propridt^ d'dtre touchy par une infinite de syst^mes de trois plans rectanglea En 
effet: le plan passant par le sommet, et perpendiculaire k la droite d'intersection de deux 
plans tangents quelconques sera un plan tangent du cdne; les plans d'un tel syst^me 
seront aussi des plans tangents de la conique mentionn^e ci-dessus: done le sommet 
du cdne sera le point d'intersection de trois plans rectangles de la conique; et ce 
sommet ^tant un point entiferement ind^termin^, le lieu de Tintersection des trois plans 
tangents rectangles de la conique, sera de mdme absolument inddtermin^, ou si Ton 
veut, ce lieu sera Tespace entier pr&s les points k une distance infinie. La contra- 
diction apparente dont M. Steiner parle, a par cons^uent son origine dans Tind^ter- 
mination qui a lieu dans le cas dont il s'agit. Dans tout autre cas, le point 
d'intersection des trois plans rectangles de la surface du second ordre est parfidtement 
ddtermin^, et les th^rfemes I. et IL sont tons deux l^gitime& 



133] 



181 



133. 



SUE UN THEORilME DE M. SCHLAFLL 



[From the Journal fQ/r die reine wnd angeivandte MathemaiUe (Crelle), torn. L. (1855), 

pp. 278—282.] 

On lit dans (§13) d'un m^moire tr^ int^ressant de M. Schlafli intituld "Uber 
die Resultante eines Systems mehrerer algebraischer Oleichungen" {M4m. de VAcad, de 
Vienne, t TV. [1852]) un tr^ beau th^r^me sur les R^sidtants. 

Pour £Eure voir plus clairement en quoi consiste ce th^reme, je prends un cas 
particulier. Soit 

F=ar» + 2)ftry + 7y" =(«, A y)(x, yy. 

Je fiEds p=a?, ? = «yi ^=y'i et je forme les op^rateurs 

81= faa + M + iCSc, 

lesquels, operant sur U, donnent 

L'op^teur 

operant sur V, donne 



182 



SUB UN THltoBilME DE M. SCHLAFU. 



[133 



Cela ^tant, soit ^==0 le rSstdUmt des ^uations U^O, F=0, c'est-iirdire I'^uation 
que Ton obtient en ^liminant w, y entre les ^uations Cr«0» F=0, ou autrement dit, 
soit ^ le resultant des fonctions U, V. Pour fixer les id^ j'^cris la valeur de ce 
r&idtant comme suit: 

^s a, Sb, Sc, d 

a, db, 3c, d 

«, 2)8, 7 
«, 2/8, 7 

Je suppose que les op^rateurs 9(, S, € op^nt sur le resultant ^, ce qui donne les 
fonctions 

a*, «*. S0, 

ou en ^rivant pour 81, S9, S leurs valeurs : 

et en consid^rant ces expressions comme des fonctions de {, 17, (T, j'en forme le 
resultant ^, savoir 

Ce resultant O contiendra le carr^ de ^ comr/ie /(icteur; c'est ce qui donne, dans le 
cas particulier dont il s'agit, le th^or^me de M. SchlaflL 

Q^n^ralement, en supposant que Ton ait autant de fonctions U, V, TT, ... que 
d'ind^termin^s x, y, ^, .••» on pent supposer que p, q, ... soient des mon6mes aftf^tf^,... 
du mSme degr^ X (il n'est pas n^cessaire d'avoir la s^rie entifere de ces mondmes), 
et on pent former des op^rateurs 9(, 93, &c. en mdme nombre que celui des mondmes 
p, f, ... avec les ind^termin^es {, 17, ... , tels que ces op^rateurs SI, S, ..., operant sur 
les fonctions 17, F, IT, ... (chacun sur la fonction It laquelle il appartient), donnent 
^(pf + 917 ...)**, ^(/>f + 917 ...)'*', &c.; t, ify &c. ^tant des mondmes de la forme x^tf^sf^ 

Cela ^tant, soit <f> le r&ultant des fonctions U, V, IT,...; en operant sur ce 
resultant ^ avec les op^rateurs 9(, 9,... et en formant ainsi les fonctions S(^, S^, ..., 
soit ^ le r&ultant de ces expressions consid^*^ comme des fonctions de {, 17, &c. 
^ contiendra une puissance de ^ comme facteur, et en supposant que /i ne soit plus 

petit qu'aucun autre des indices /*, /*',...; 7r = /A/i'...; et o-= - + —> + ..., I'indice de 
cette puissance sera au moins a . Voil^ le th^rfeme g^n^ral de M. SchUlflL 



138] 8UR UN THiOR^ME DE M. SCHLAFLI. 188 

La demonstration donn^e dans le m^moire cit^ est, on ne pent plu8» simple et 

Aigsjite. E!Ue repose d'abord sur un th^r^me connu (d&nontr^ au reste § 6) qui 

pout dtre ^nonc^ ainsi; savoir, en supposant que les Equations {7=0, F&=0, ... soient 
satisfisdtes, on aura (prte un facteur ind^pendant de {, 17,...) ^ 

^<l>^t(p^ + qfj...y, 8^ = ^ (pf + 517. ..>*', &C. 

Puis, elle est fond^ sur le th^rfeme d^montr^ (§ 12), savoir: le r&ultant des 
fonctions 






{oh /, /*,... sont des poljnidmes de degrds /*, /*',... en f, 17, &c., et p, g, ..., t, ^, ... 
des constantes quelconques) sera, en supposant que /i ne soit plus petit qu'aucun autre 

des indices fi, M^•••> et en posant 7r=/i/i'..., tout au plus du degr^ - par rapport 

aux quantity t, If, &a Voici cette demonstration, qui suppose aussi que le r&ultant 
^ soit indecomposable, Supposons que les coefficients de U, F, IT, ... soient assujettis 
k la seule condition d'etre tels que le r&ultant ^ soit un infiniment petit du premier 
ordre, il sera permis de supposer que tous ces coefficients des ind^termin^ x, y, ... 
ne different des valeurs qui satisfont aux ^nations U^O, F=0, TT^O, ... que par des 
increments infiniment petits du premier ordre; le resultant if> sera un infiniment petit 
da premier ordre, mais toute autre fonction des coefficients, k moins qu'elle ne contienne 
nne puissance de ^ comme fiekcteur, aura une valeur finie, et toute fonction des 
coefficients infiniment petite de I'ordre k contiendra ^ comme facteur. Dans cette 
supposition les Equations S[^ = 0, S^ » 0, &c. deviendront : 

oil/ /',..> sont des polyndmes de degr^s /li, fi\ ... dont les coefficients sont des 
infiniment petits du premier ordre. En supposant toujours que /* ne soit plus petit 

qu'aacun autre des indices /i, /i^... et en posant 7r = fifi\.., o- — — + ->...> le resultant ^ 

fAf fit 

du syst^me sera tout au plus du degrd — par rapport aux quantit^s finies t, t^ .... Le 
d^ par rapport k tous les coefficients est o-; le degr^ par rapport aux coefficients 
<le /,/',... sera done au moins o- ; c'est-Jrdire, ce resultant sera un infiniment 

petit de I'ordre o- , ou enfin, ^ contiendra ^^"^'f^ comme facteur. Or les coefficients 

^ U, F, Wf... (assujettis k la seule condition ci-dessus mentionn^e) etant d'ailleurs 

^Mtraires, on voit sans peine qu'il est permis de &ire abstraction de la condition, et 

que <t contiendra en gdn^ral cette mdme puissance ^^^-^'i^ comme facteur; ce qu'il 
^agisBait de d^montrer. 



184 SUB UN THltoBilME DE M. SCHLAFLL [l33 



Rien n'emp^he que ^ ne contienne une plus haute puissance que ^^^^^-i^ comme 
facteur, ou que ^ ne s'^vanouisse identiquement. On pent mdme assigner de plus 
pr^ que Fa (ait M. Schlafli, des cas o{i ^ s'^vanouit identiquement Soient m, m\ m'\ . . . 

les degr& de U, V, W,.,. par rapport i, a?, y, 2r,... , p = mmW ...,« = — + —,+ ^, ... , 



771 771 771 



if> sera du degr^ ^ par rapport aux coefficients de U. Soient aussi fi^, fi,,, ... les 
degr^s de celles des fonctions S(^, 9^, . . . , pour lesquelles les op^rateurs S[, S, . . . 
contiennent des diff^rentielles par rapport aux coefficients de U, p = — I — ... : pour 

ces fonctions les coefficients seront du degr6 — — 1 par rapport aux coefficients de U ; pour 

les autres ils seront du depr^ — . 4> sera done du degr^ (~ — l)p + ^(o- — p), = — o- — p, 

m \fn J m m 

par rapport aux coefficients d^ U, et ^-r^'^-'* sera du degr^ —a — p"^ (a j, c'est- 

k-diie du degrd *- . p par rapport aux coefficients de U, De m^me, en supposant 



que les lettres m', p', ... aient rapport It V, &c., O-s-^'^''* sera du degrd —,. p\ &c. 

lit Uf 

par rapport aux coefficients de F, &c. Si Tun quelconque des nombres — . — p, 
^, . p', &c. est nigatif, et It plus forte raison, si leur somme 8 . — o- est negative^ 

771 Uf Ut 

^ doit s'^vanouir identiquement. En particulier, en supposant que le nombre des 
fonctions S[^, S^, ... (c'est-lt-dire le nombre des ind^termin^ ^, 17, ...) soit v^ on aura 

a->v- , et par cette ndson ^ s'dvanouira identiquement si - (o-— i/) est n^gatif, c'est- 

It-dire si v>a. Je ne parlerai pas ici des cas examine par M. Schlaffi, oil 4> con- 
tient comme facteur une plus haute puissance que ff^r^'i^. 



134] 



185 



134 



KEMAEQUES SUE LA NOTATION DES FONCTIONS 

ALGEBEIQUES. 



[From the Journal fWr die reine und angewandte McUhematik (CSrelle), torn. L. (1855), 

pp. 282—285.] 



Je me sers de la notation 



«', fi", y\ 

of\ P\ i\ 



pour repr^nter ce que j'appelle une matrice; savoir un systime de quantit6s rangdes 
en forme de carrS, mais d'ailleurs tout It fait indSpendantes (je ne parle pas ici des 
matrices rectangulairea). Cette notation me parait tr^s commode pour la th^rie des 
^luations UnAiirea; j'^ris par exemple 



(f, V» ?, ...) = ( 



« * fi 9 7 > ••• 

/»" iO" -/' 



X«^> y» «> •••) 



pour repr^nter le syst^me des ^nations 

f = « x-^-fi y-^ry z ... , 

c. n. 



24 



186 



REMABQUES SUB LA NOTATION DE8 FONCTIONS ALG^BRIQUES. 



[134 



On obtient par Ik T^quation: 

(a?, y, z, ...) = ( 



a, /3 , 7 , ... r^Xfi Vf (T, 
a', 13'. y\ ... 

Q^'i ^', y, ... 



••/» 



qui repr^nte le syst^me d'^uations qui donne x, y, z, ... en termes de {,17, (r> *.• > 
et on se trouve ainsi conduit k la notation 



— 1 



« I ^ > 7 > ••• 

*'f ^'» 7'» ••• 
-,// jcy^ ^// 



de la matrice tnt^er«6. Les termes de cette matrice sont des fractions, ayant pour 
d^nominateur commun le determinant formd avec les termes de la matrice originale; 
les numiriiteurs sont les determinants mineurs form6s avec les termes de cette mSme 
matrice en supprimant Tune quelconque des lignes et Tune quelconque des colonnes. 



Soit encore 



(a?, y, z, ...) = ( 



tty Oy Cy... 

_/ r/ -/ 

tVy " t Cy... 

_// r// -// 

Cvy ^> Cy... 



Xa?, y, z, ...), 



on pent ^rire: 



.) = ( 



a', 



^ , 7 1 ••• 




/s*. y.... 




/9". 7".... 





a', 
a", 



Of C f • .. 

Of C y . . . 

Of V y • . • 



A^f Jft Zf ...}, 



et Ton parvient ainsi k Yid6e d'une matrice compoaie, par ex. 



a , )8 , 7 , ... 

«'» ^'> y* ••• 

/»'' iO'' «,/' 

« > P , 7 > ••• 



Cvy ^> Cy... 

Cvy ^1 C f • • . 

d'f Vf (j'f ... 



On voit d'abord que la valeur de cette matrice compost est 

(«» fi> 7> •••X^» ^'» ^"* •••)» (*» ^» 7> •••X^> 6'i 6''* •••)> ••• 



134] 



BEICABQUES SUB LA NOTATION DBS FONCTIONS ALGJ^BIQUES. 



187 



oil (a, fi, 7, ...Xa, o,\ a"i ...) = «« + i8a' + 7a'' + ... . II faut fiaire attention, dans la com- 
position des matrices, de combiner les lignes de la matrice It gauche avec les cclonnes 
de la matrice It droite, pour former les lignes de la matrice compost. II y aurait 
bien des choses It dire sur cette th^orie de matrices, laquelle doit, il me semble, 
pr^c^er la th^rie de Diterminants, 

Une notation semblable pent dtre employee dans la th^rie des fonctions quad- 
ratiques. En effet, on pent d^noter par 



( 



a , fi , 7 , ... 
*'> ^» Vf ••• 

M^' /y ^." 



Xf. % fXa?, y, ^) 



la fonction linwhlvniaire 



et de lib par 



la fonction qwxdraiiqaj^ 






( 



a, A, ^r, . 
A, 6, /, . 
5^, /, c,.. 



X«> y. -8:, ...)» 



oa^ + 6y* + c-?* + 2/y2r + 25r-^a? + 2Aa;y ... 



que je repr^nte aussi par 

(a, 6, c, .../, g, h, ...X^, y, -^f ...)". 

Je remarque qu'en g^n^ral je repr^sente une fonction rationnelle et int^grale, 
homog^ne et des degris m, m\ &a, par rapport aux ind^termin^es x, y, &c., a/, y', &c., 
de la mani^re suivante: 

(0X^» y,'"T{^y y', ...)"•' .... 

Une fonction rationnelle et int^grale, homog^ne et du degr^ m par rapport aux deux 
ind^termin^ d?, y sera done repr^nt^e par 



( X^, yT' 



24—2 



188 BEMABQUES SUB LA NOTATION DES FONCTIONS ALQ]£!BBIQUES. [l34 

En intFoduisant dans cette notation les coefficients, j'^ris par exemple 

(a, 6, c, d$x, yy, 
pour repr^nter la fonction 

tandis que je me sers de la notation 

(a, 6, c, cCJlx, yy, 

pour repr^nter la fonction 

(M5* + ba^ + cay* + dy*, 

et de mSme pour les fonctions d'un degr^ quelconqua tTai trouvd cette distinction 
tr^ commode. 



[In the foregoing Paper as here printed, except in the expression in the second line of this page, 
)( is nsed instead of y{ : it appears by a remark {CreUe, t. u, errata) that the mannsoript had the inter- 
laced parentheses J^ . Moreover in the manuscript ( ) was osed for a BCatrix, which was thus distingnished 

from a Determinant, but in the absence of any real ambiguity, no alteration has been made in this respect. 
In the reprint of subsequent papers from Grelle, the arrowhead }( ^^^ 3[ ^ ^^'^ instead of {J) . ] 



135] 



189 



135. 



NOTE SUR LES C0VARIANT8 D'UNE FONCTION QUADRATIQUE, 
CUBIQUE, OU BIQUADRATIQUE A DEUX INDETERMINEES. 



[From the Journal filr die reine und angewandte Mathematik (Crelle), torn. L. (1855), 

pp. 285—287.] 

La th^rie d'une fonction k deux ind^termin^es d'un degr6 quelconque, par example 

( X^, yn 

depend du syst^me des cavaricmta de la fonction, lequel est cens^ contenir la fonction 
elle-mSme. 

Pour une fonction quadratiqiie le syst^me de covariants est 

(a, 6, cXa?, yA 
ac — 6*. 

Pour la fonction cubiqtie, le systfeme est 

(a, 6, c, dX^f, y)», 

(ac — h^, ad — bc, hd — (^){x, y)", 

(- a« + 3a6c - 26", -aM+2ac*-6^, ood-26« + 6c», ad>-3&cd + 2c»)(a?, y)\ 

- a*cP + 6a6cd - 4ac« - 46»d + 36»(J«, 

lonctions lesquelles, en supposant qu'on les reprdsente par U, J7, ^, Q, satisfont iden- 
^^ement It I'&iuation 



190 



NOTE SUB LES COVAMANTS DUNE FONCTION QUADRATIQUE, 



[135 



Pour la fonction biqiuidratique, le syst^me est 

(a, 6, c, d, cX«?, y)*, 

(ac-b', 2ad-2bc, ae + 2bd-3(f, 266-2cd, ce-d?Xx, y)*, 

ace + 26cd — ocP — l^e — c*, 

/^ - a»d + Sabc - 26», 

-a»c -2a6d +900" -66*c, 

-5a66 +15acd-106^, 

+ lOa'd - 106>6, I (a?, y)», 

+ 6ad6 + 106(? - 155ce, 

+ a6» +26<fo -9c»c +6c(P, 

^ +6e* -Sccfe +2d», 

et ces fonctions, en supposant qu'on les repr^nte par U, /, J7, /, O, satdsfont iden- 
tiquement It I'^quation 



J'ajoute k ce systfeme la fonction 

/» - 27/» = aV - 1 2a»6(fo» 
+ 54a6*c6* - 6a6»d«6 
-54ac»d« -276V 
+ 366»C\P, 



ISaVc* +54a%eP6 -27a^ 
180a6c'c26 + lOSabcd^ + 81ac% 
646»d« + lOSb'cde - 646Vc 



qui est le discrimincmt de la fonction biquadratique. 

Pour donner une application de ces formules, soit proposd de r&oudre une ^nation 
quadratique, cubique ou biquadratique, ou autrement dit : de trouver un facteur liniaire 
de la fonction quadratique, cubique, ou biquadratique. 

U est assez singulier que pour la fonction quadratique la solution est en quelque 
sorte plus compliqu^e que pour les deux autrea En effet, il n'existe pas de solution 
sym^trique, It moins qu'on n'introduise des quantit^s arbitraires et superflues; savoir, 
on trouve pour facteur lin^aire de (a, 6, cX^, y)* Texpression 

(a, 6, cXa, iSXa?, y) + V- D .(i8«-ay), 
oil (a, 6, cX«, /8Xa?, y) denote aaa + b (ay + fix) + c/Sy. 

Pour la fonction cubiqrie, T^quation O* + D iT'* = — 4tH* fisdt voir que les deux fonc- 
tions 4> + I7V — D , ^ — ^V — D soiit Tune et Tautre des cubes parfedts. L'expression 

V {H^ + u-/ - o)} - y [H^ - u^ - m 



135] CUBIQUE, OU BIQUADRATIQUE A DEUX IND^TERMHrflES. 191 

sera done une fonction lin^aire de x, y; et puisque cette fonction s'dvanouit pour 
U=0, elle ne sera autre chose que Tun des facteurs lin&dres de (a, b, c, d)(x, yy. 

Pour la fonction biquadrabiqtie, en partant de T^quation 
j'^cris 

et je mets T^uation sous la forme 

(1, 0,-Jf, M)(IH, JUy=^-iP^. 

Done, en supposant que tJi, vr^, «j, soient les racines de T^quation 

(1, 0, - M, MXm, If = 0, 

OU plus simplement de T^uation 

t!r»-Jlf(t!r-l) = 0, 

ces expressions IH — vtiJU, IH — vrJU^ IH — vr^JU seront toutes trois des earr& de 
fonctions quadratiquea L'expression 

sera done une fonction quadratique, et on voit sans peine qu'elle sera le carr^ d'une 
fonction liniaire. Or cette expression s'^vanouit pour {7=0; done ce sera pr^eis^ment 
le earrd de I'un quelconque des facteurs lin^aires de (a, 6, c, d, e){x, yf. 

L'^uation identique pour les covariants d'une fonction biquadratique donne lieu 
aussi (remarque que je dois k M. Hermite) It une transformation tr^s ^l^gante de 



tinUgrdU dliptique |da?-^V(a, 6, c, d, e^x, 1)*. 



192 



[136 



136. 



SUE LA TKANSFOKMATION D'UNE FONCTION QUADRATIQUE 
EN ELLE-M^ME PAR DE8 SUBSTITUTIONS UNfiAIRES. 



[From the Journal fWr die reine und angewandte Mathemoitik (Crelle), torn. L. (1855), 

pp. 288—299.] 



Il s'agit de trouver les transformations lin^aires d'une fonction quadratique 
( )(«! y, ^» •••)* ^ eUe-mSme, c'est-lt-dire de trouver pour («, y, z, ...) des fonctions 
lin^aires de x, y, z, ... telles que 



En repr^ntant la fonction quadratique par 



(0X^» y» ^^ •••)* = 



o. 


A. 


9* .•• 


h. 


b, 


J 9 ••• 


9. 

• 

• 
• 


/. 


c , • • • 



Xa?, y, -er, ...)«, 



la solution qu'a donn^e M. Hermite de ce probl^me peut dtre rdsum^ dans la seule 

&|uation 

(x, y, z,...) = 



( 



a, h,g, ... 
K b,f, ... 
fft Jf ^f ••• 



— 1 



a, h-p, g + fi, ... 
h + v, 6, /— ^ ••• 



o, h + v, g — fi, ... 
h — p, b, f+\ ... 



— 1 



a, h,g, ... 
A, 6,/, ... 

gf j$ ^f • • • 



)\*^ty»^$»")» 



oil X, /i, I', ... sont des quantit^s quelconques. 



136] SUB LA TRANSFOBMATION d'uNE FONCTION QUADRATIQUE &C. 193 



En effet, pour ddmontrer que cela est une solution, on n'a qu'^ reproduire dans 
un ordre inverse le proc^d de M. Hermite. En introduisant les quantity auxiliaires 
(i> V» K* •••)> ^^ P^^^ remplacer T^uation par les deux Equations 



( 



( 



a, A, g, ... 

A, 6, /, ... 

9* ji c, ... 

a, A, ^r, . . . 

K ft, /, ... 

5^1 ji c, ... 



Xa?, y, z, ...) = ( 



A^> y» ^> ••./ — \ 



a, A + i/, g — fi,... 

A — v, 6, y + x, ... 

* 

a. A -I/, ^r + A*, ... 

A + ^'i &> / — X, . . . 



Xf» ^» ?» •••) 



Xf» ^» ?» •••) 



qui donnent tout de suite d'abord 

et puis 

a: + x = 2f, y + y = 2i;, ^ + z = 2j; &c. 

On obtient par Ul: 

(OXx, y, z, ...)» = ( 0X2f-x, 2i;-y, 2?- z, ...)», 

=4(oxf. ^. ?, ...)»-4(oxf, ^> r....x^, y. -^^ ...) 

c'est-il-dire T^uation 

( Xx> y. z, ...)»=( X^'» y. -^^ •••)"» 

qu'il s'agissait de verifier. 

Je remarque que la transformation est toujours propre. En effet, le determinant 
de transformation est 



a, h, g ... 
A, b, f ... 
9,f, c ... 



—I 



h+v, b, y— X ... 

g-fi,f+\ c ... 



Oi h + v, g — fi ... 
h — v, b, y + X ... 
fl^ + A*, /-X, c ... 



— 1 



a, A, ^r ... 
A, 6, / ... 
g> f, ... 



Or les determinants qui entrent dans les deux termes moyens, ne contiennent Tun 
ou Tautre que les puissances paires de X, fi, v, ... . Done ces deux determinants sont 
^ux, et les quatre termes du produit sont rdciproques deux k deux; le determinant 
de transformation est done + 1, et la transformation est propra 

Pour obtenir une transformation improprey il faut consid^rer une fonction quadratique 
qui contient outre les indeterminees x, y, z, ... une indeterminee 0, et puis r^duire k 
C. II. 25 



194 



SUB LA TRANSFORMATION d'uNE FONCTION QUADRATIQUB 



[136 



z^ro les coefficients de tous les termea dans lesquels entre cette ind^termin^ 0. Les 
valeurs de z, y, z, ... ne conidendront pas 0, et en repr^ntant par % rinddtermin^ 
que Ton doit ajouter k la suite z, y, z, ... , la valeur de % sera, comme on voit sans peine, 
^ = — 0; le determinant de transformation pour la forme auz inddtermin^ w, y, z, ,..y0 
sera + 1| et ce determinant sera le produit du determinant de transformation pour la 
forme aux ind^terminees x, y, z,.., multipli^ par —1. Le determinant de transformation 
pour la forme aux indeterminees x, y, z, ... sera done —1, c'est-jl-dire, la transformation 
sera impropre. 

Au lieu de la formule de transformation ci-dessus, on peut se servir des formules 



(f V> ?, ••) = ( 



h^p, b , y+x,... 



> ••• 



— 1 



a, A, jr,... 
9* J* c,... 



X*> y» *> •••)» 



x = 2f-a?, y = 2i;-y, z=2f-2r, .... 
Par exemple, en supposant que la forme k transformer soit 



on aura 



(f % t •••) = ( 



a, I/, -A*,... 

— I/, 6, X, . . . 

M> — ' A», c, . . . 



•%ax, by, cz, ...), 



x«2f-a:, y = 2ty-y, z = 2f--r, Ac, 



de mani^re qu'en posant 



a, v, -^,... 
■ i/, 6, \, . . . 
M> X, c, . . . 



=*, 



oil aura 



(x, y, z •••)— f 





a, V, 


-/*«••• 


-.-1 


-.'. 6, 


A>) ... 




• 
• 
• 


C| • • . 




1 

it 


a 


2£ 


f 






2A' 



2^ 



// 



25'-- 
6' 



25" 



2C 



2C 



2(7"--... 



Xcue, by, cz, ...), 



136] EN elle-m£mb par des substitutions lin^ires. 195 

ce qui est I'^uation pour la transformation propre en elle-m^me, de la fonetion 
a^ + &y*-f cei* + &K^ On en d^uira, comme dans le cas gdndral, la formule pour la 
transformation impropre. On trouvera des observations sur eette formule dans le 
m^moire "Becherches ult^rieures sur les determinants gauches" [137]. 

Je reviens k T^uation gdn^rale 

( Xx, y, z, ...)» = ( x^. y. ^. •••)*. 

et je suppose seulement que x, y, z, ... soient des fonctions lin&dres de x, y, z, ... qui 
satisfont k cette ^nation sans supposer rien davantage par rapport k la forme de 
la solution. Cela ^tant, je forme les fonctions lindaires z — sx, y — sy, z — sz, &c., oil s 
est una quantity quelconque, et je consid^re la fonetion 

(OXx--«a:, y-«y, z-«^ ...)», 
laquelle, en la d^veloppant, devient 

(i + «*XO)(^, y, z...y^2s(<>){x, y, Z...XI % r...); 

et en d^veloppant de la mSme mani^re la fonetion quadratique 

(0)(x--ar, y-jy, z--^, ...j , 

on obtient I'^uation identique 

/ 1 1 1 \« 

( OXx-«a?, y-«y, z-«^, ...)» = «».( 0)(x--a?, y--y, ^"g^'"')' 

Soit n 1^ determinant formd avee les coefficients de fonctions lin^ires x — sx, 
y-sy, z— «z, &c. En supposant que le nombre des inddtermin^es x, y, z, &c., est n, 
sera ^videmment une fonetion rationnelle et int^grale du degrd n par rapport k 8. 
Soit de mSme Q' 1^ determinant form^ avec les coefficients de 

1 1 1 . 

x--ir, y--y, z--^> &c.; 

r^uation qui vient d'etre trouv^e, donne □« = «** □'«, c'est-k-dire Q = ± «* D'- Cela 
&it voir que les coefficients du premier et du dernier terme, du second et de Tavant- 
dernier terme, &c., sont dgaux, aux signes pr&& De plus, le coefficient de la plus 
haute puissance «* est toujours ± 1» et on voit sans peine qu'en supposant d'abord 
que n soit impair^ on a pour la transformation propre: 

n=(i, p,...p, ix-«. ir 

et pour la transformation impropre 

n = (l, -P....P, -lX-^> If: 

Equation qui pent 6tre chang^ en celle-ci: n = — (1> P, ... P, 1X*> I)**- P^^» ^^ 
supposant que n soit pair, on a pour la transformation propre: 

n=(i, P, ...p, ix-«, 1)^ 

et pour la transformation impropre: 

n=(i, -P....P, -ix-«» m 

25—2 



i 



196 SUR LA TRANSFORMATION d'uNE FONCTION QUADRATIQUE [136 

le coefficient moyen dtant dans ce cas 4geA k z6ro, Ces th^rfemes pour la forme du 
determinant des fonctions lin^aires x — «r, y — «y, z — «z,' . . . sent dus k M. Hermite. 

II y a ^ remarquer que la forme ( X^> y» z ...y est tout k Mi inddterminfe; 
c'est-^-dire, on suppose seulement que x, y, z, ... soient des fonctions lin^aires de 
^> y» ^, •••> telles qu'il y ait une forme quadratique ( X^» y> ^> •••)* P<>^r laquelle 
r^uation ( X^, y, z ...)^ = ( X^> y, ^ ^-Y ^t satisfaite. 

Je regarde d'un autre point de vue ce probl^me de la transformation en elle- 
mSme, d'une fonction quadratique par des substitutions lin^ires. Je suppose que 
X, y, z, &c. soient des fonctions lin^aires donn^es de x, y, z, ... , et je cherche une 
fonction lin^aire de x, y, z, &c qui, par la substitution de x, y, z, &c. au lieu de 
X, y, z, &c. se'transforme en elle-mSme k un facteur prfea Soit (f, m, w, ...X^, y, z, ...), 
cette fonction lin^ire, il faut que (f, m, 7i, ...Xx, y, z, ...) soit identiquement =«.(f, m, w, ...) 
(a?, y, z, ...), ou, ce qui est la mfime chose, que (f, m, n, ...Xx — «p, y — «y, z — «^, ...) soit = ; 
c'est-ji-dire, les quantity Z, m, n, ... seront ddtermin^es par autant d'^uations lin^aires 
dont les coefficients sont pr^is^ment ceux de x — «a?, y — sy, z — sz, &c. ; done s sera 
d^termind si Ton rend dgal i zdro le determinant formd avec ces coefficients, et Z, m, n, &c. 
86 trouveront donnas rationnellement en termes de 8. Cela ^tant, je suppose que les 
racines de T^quation en 8 soient a, 6, c, . . . , et ces diffdrentes racines correspondront 
aux fonctions lin&ires x^, x^, x^., ... qui ont la propri^te dont il s'agit. Soit ( X^» V* ^* •••)' 
une fonction quadratique qui se transforme en elle-m^me par la substitution de x, y, z, &c. 
au lieu de a?, y, z, &c. Cette fonction pent 6tre exprim^e en fonction quadratique de 
Xa, Xft, x«, &C.; quantitds qui, en substituant x, y, z, &c. au lieu de x, y, z, ... deviennent 

(t^H y 0X5 ) CX^ , . • » . 

Je prends les cas d'une fonction binairef temairej &c., et d'abord le cas d'une 
fonction binaire. 

En ^crivant d*abord ( X^> yY^i-^t ^» C^X^a> x^)*, on doit obtenir identique- 
ment (A, B, C) (ax«, 6xft)« = (il, B, C)(Xa, x^y, c'est-i-dire ^(a«-l) = 0, 5(a6-l) = 0, 
(7(6*— 1) = 0. Or la solution A=B = C = ne signifiant rien, on ne pent satisfaire k 
ces Equations sans supposer des relations entre les quantit^s a, 6; et pour obtenir une 
solution dans laquelle la fonction quadratique ne se rdduit pas k un carre, il £Etut 
supposer, ou aft — 1 = 0, ou a' — 1 = et 6" — 1 = 0. Le premier cas est celui de la 
transforination propre. II donne 

a6 = l, ( 0X^» y)» = ZxaX6. 

Le second cas est celui de la transformation impropre. II donne 

a = + l, 6 = -l, ( OX^* y)' = ^V-l-wXft«. 

En passant au cas dune fonction temaire, soit 

( X^, y» ^y^(^. B, C, P, (?, jyXxa, Xft, x,)«i 
on doit avoir identiquement 

{A, B, C, F, G, HXaXa, 6x,, cx,)« = (il, «, C, F, (?, ^x„, x,, x,)«. 



136] EN ELLE-mMe par DES substitutions LINilAIRES. 197 

c'est-il.dire^(a«-l) = 0, £(6>-l) = 0, C(c»-1)=0, J^(6c-1) = 0, (?(ca-l) = 0, Hiah-l) 
= 0, et on voit que pour obtenir une solution dans laquelle la fonction quadratique ne 
sse r^uit pas k un carrd» ou k une fonction de deux ind^termin^es, il faut supposer 
par exemple a* — 1 = 0, 6c — 1 = 0. On a done dans le cas d'une fonction temaire : 

a'=l, 6c = 1, ( X^> y» '2:)» = Zxa' + mx6Xe. 
Xa transformation sera propre, ou impropre, selon que a = + l ou a = — 1. 

Dans le cas d'une fonction qtuitemaire, on obtient pour la transformation propre: 

oi = cd = 1, (<>)(a!y y, z, wy = l XaXb + m XcX^, 
est pour la transformation impropre: 

a = + l, 6 = -l, crf = l, ( X^> y. -^^ ^)' = ^ V + mxfr« + nXcXd. 

Dans le cas d'une fonction quinaire on obtient 

a*=l, 6c = dc=l, ( X^> y» ^» ^> t)' = Zx«« + mxftXc + wxrfX« 

est la transformation est propre ou impropre, selon que a = + 1 ou a = — 1 ; et ainsi 
de suite. 

Cette mtStbode a des difScultes dans le cas oil T^uation en ^ a des racines 
egales. Je n'entre pas ici dans ce sujet. 

Dans les formules qu'on vient de trouver, on pent consid^rer les coefficients 
£, m, isQ, coqime des quantit^s arbitraires. Mais en supposant que la fonction quadra- 
t.ique soit donnie, ces coefficients deviennent dAerminis. On les trouvera par la formule 
»uivante que je ne m'arrSte pas k d^montrer. 

Soient a, /S, 7, &c. les coefficients de la fonction lin^ire x^, a', P\ 7', &c. les 
ooefficients de la fonction lin^aire z*, et ainsi de suite; alors, dans les diffi^rentes 
fiormules qui viennent d'etre donnas, le coefficient d'un terme Xa' k droite sera 

-it 



et le coefficient d'un terme XaX^ k gauche sera 



(tX«» P* 7. •••X<»'> ^> 7'> •••)' 



oi!i k denote le discriminant de la fonction quadratique k gauche, et ou les coefficients 
des fonctions quadratiques des d^nominateurs sont les coefficients inverses de cette mSme 
foiiction quadratique k gauche ^ 

' Je profite de oette oooaaion pour remarqner oonoemftnt oes reoherches que les fonnules donn^es dans 
^ note stir lee foxictioiiB dn seoond ordre (t zumi. [1S4S] p. 105) [71] poor les cas de trois et de qoatre 
^^tennin^es, sont ezaolea, mais que je m'^tais iromp6 dans la fonne g^n^rale dn thtor&me. [This correction 
U indicated toI. x. p. 589.] 



198 SUR LA TRANSFORMATION d'UNE FONCTION QUADRATIQUE [136 

L'application de la mdthode k la forme binaire (a, b, c){m, yY donne lieu aux 
d^veloppements suivants. 

J*&;ris x=aa? + /8y, y=7« + Sy, et je repr^nte par (I, m)(x, y) ime fonetion 
lin^aire qui par cette substitution est transform^e en elle-mSme, au &cteur 8 pres. 
Nous aurons done 

{I, mXax + py, ya + Sy)^8 (I, m){x, y) ; 

r^uation pour 8 sera 

«»-«(S + a) + aS-)87 = 0; 

laquelle pent aussi 6tre ^rite comme suit: 

(1, -S-a, aS-/37X«, 1)' = 0. 

Soient 8\ tt' les racines de cette ^nation. (II est k peine n^cessaire de remarquer 
que «', 8"y et plus bas P, Q, sont ici ce que dans les formules gdn^rales j'ai repr^- 
sentd par a, 6 et Xa, x^. De mSme les Aquations p = «'«", p^8'\ p^8"\ obtenues 
apr^y correspondent aux ^nations a& = l, a^ = l, &* = !.) On aura 

«' + «" = - S-o, «V' = aS-i87, 
et les coefficients 2, m seront ddtennin^ rationnellement par s, 

Mais on pent aussi determiner ces coefficients par I'^uation 

I : m = ia + m7 : 1/3 +mB, 

qui pent etre ^crite sous la forme 

(/3, S-a, -.7X/, m)« = 0. 

et en ^liminant entre cette ^nation et Tdquation lx + my = les quantity I, m, on 
voit que les fonctions lin^aires Ix + my sont les facteurs de la fonetion quadratique 
(/3, S — a, — 7)(y, — a?)", ou, ce qui est la m^me chose, de la fonetion quadratique 

je repr^nte ces facteurs par P, Q et je remarque encore que T^uation en s aura 
des racines ^gales si 

(S-a)» + 4)87 = 0, 

et que dans ce cas, et exdttsivement dans ce cas, les fonctions P, Q ne forment qu'une 
seule et mSme fonetion lin^aire. 

Je suppose maintenant que la fonetion (a, b, c)(x, yY se transforme en elle-mSme 
par la substitution aa-{-fiy, yx-^By au lieu de a?, y, ou, ce qui est ici plus commode, 
je suppose que les deux fonctions sont ^gales k un &cteur prte, et j'^ris 

(a, 5, cXouv + fiy, 7a? +8y)» = p (a, 6, c)(x, yy. 



136] EN ELLE-M^E PAB DES SUBSTITUTIONS LIN]6 AIRES. 199 

En d^veloppant cette Equation, on obtient 

a;» (a, 6, cXq? - p, 2a7, 7» )^ 

+ 2 a:y (a, 6, cXa/3, aS + fiy-p, yS )^=0. 

+ y*(a, 6, cXy8«, 2/3S, S«-p), 

VoiUl trois ^nations lin^aires pour determiner par les quantitds a, /3, 7, S, consid^r^es 

comme donn^, les coefficients (a, b, c) de la fonction quadratique. Les coefficients de 

ces Equations lin^aires sont 

a«-p, 207, 7», 

flf/9, aS + fiy- p, 78, 

i8», 2/38, 8«-p. 

Le syst^me inverse par lequel on trouve les valours de a, 6, c, est 

«»(aS-/87)-(aS + i97 + 8")p-l-p», - jSS (aS - /87) + fl^p, 
- 27* (oS - /87) + 2a7p. a«8« - )8V - (S* + «») p + p*, 

7* (aS — ^87) + 7"p, — 07 (aS — /97) + 7Sp, 

fi^(ab^fiy)^/3»p, 
- 2a^ (oS - /97) + 2i8Sp, 

a«(flfS-iS7)-(aS + /87 + a«)p + p», 
et le determinant, dgal^ k z6ro, donne 

(aS-)87-p){(aS-/97 + p)«-p(a + S)»}=0: 
^uation dont les racines sont 

p = aS-/97, p={i(a + S)±iV(a-S)> + 4)87}>. 

En comparant ces valours avec cellos de s', 8'\ on voit que les racines de T^uatiou 

en p sont 

P = «V', p = «'S p = A 

et nous aliens voir que ces valours de p donneut en gdndral les valours F(l^ P*, Q*, 
pour la fonction quadratique. 

Soit d'abord p = aS — )87 (= ^V), et posons pour abr^ger oS — )87 — p = ^, le systfeme 
inverse devient: 

(S'-p)*-/8p. 27, -/3S^-i9p(S-a), ^<l> + /3p,2fi, 

-2780-p(S-a)27, (a8 + ^y-p)<^- p(S-a)», - 2a^<^ + p (S - a) 2/S, 

-7»^ + 7p.27, -a7<^ + 7p(S-a), (a»-p)^-7p . 2/8, 

et en mettant ^==0, les termes de chaque ligne (en omettant un fisK^teur) deviennent 
7, ^ (5 — a), fi. On obtient ainsi dans ce cas, pour la fonction quadratique (a, b, c){x, yY 
la valour 

(% s-«, -0X^,yy> 

qui est en effet le produit PQ des fonctions lin^aires. 

n y a ^ remarquer qu'en supposant (S — a)* + 4^87 = 0, ce qui est le cas pour lequel 
p sera une racine triple, il n'y aura pas de changement k hire dans ce r&ultat. La 
fonction quadratique est, comme auparavant, le produit PQ des fonctions lin^aires; 



200 SUR LA TRANSFORMATION d'UNE FONCTION QUADRATIQUE [136 

seulement ces deux fonctions lin^ires dans le cas actuel sont identiques, de mani^re 
que la fonction quadratique se r^uit k P*. 

Soit ensuite 

p = {i(a + S)±iV(a-«)» + 4i87}H=«'* ou «"»); 

en ^rivant p = «■ et en mettant pour abrdger a8 — ^87 — « (S + a) + «• = x» 1® systfeme 
inverse devient 

- 2iyBx - 27J? (S - 8) (S + a), {«» + « (8 + a) + aS + ffy} x + 2l3y8 (8 + a), 

7^ + 7"« (8 + a), - OTx - 7« (« - «) (8 + a), 

/8»x + /8»« (S + a), 

Done, en ^rivant x = ^ ^^ ^^ omettant le fisK^teur 8{S + a), le syst^me inverse devient 

et les quantity dans chaque ligne sont dans le rapport l^ : hn : m*, de manifere que la 
fonction quadratique est dans ee cas dgale k P^ on Q^. Cela suppose que S + s ne soit 
^gal k zdro. En faisant pour le moment p = 1, on en tire la conclusion qu'd. moins 
de supposer 8 + a = 0, il n'existe pas de fonction quadratique binaire proprement dite 
(fonction non carr^e) qui par la substitution impropre ax-^-fiy, 7^ + Sy pour x, y, se 
transforme en elle-mSme. L'^uation S-f a = donne p = aS — /97, qui est une racine 
double de Tdquation cubique. On remarquera en passant par rapport k la signification 
de r^uation S + a = 0, que Ton a en gdndral: 

(a, /8Xap + i9y, yx + Sy) : (7, SXflw? + i9y, ya^ + By) 
= (a« + i97)^ + )8(S + a)y : y(B-h a)x-{-(S' + I3y)y, 

et de Ik, qu'en supposant 8 + a = 0, on a 

(a, fiXax-Vfiy, yx + Sy) : (7, BX^x-^fiy, yx+Sy) = x, y. 

Cela revient k dire qu'en faisant deux fois la substitution ax + /3y, yx + Sy au lieu de 
X, y, on retrouve les quantity x, y, ou que la substitution est pSriodique du second 
ordre. II y a aussi k remarquer que dans le cas dont il s'agit, savoir pour S + a = 0, 
on a 8" = —8\ et que les deux fonctions lin^aires P, Q restent parfaitement dc^termindes. 

Nous venons de voir qu'il n'existe pas de transformation impropre d'une fonction 
quadratique binaire proprement dite, k moins que S + a ne soit pas = 0. Mais en 
supposant S + a = 0, on voit que les coefficients des ^nations pour a, b, c deviennent 

-fir -7(«-«X 7'> 
a/9, a(S-a), -a7, 

/8«, i9(8-a), fiy, 



136] EN ELLE-M^ME PAR DES SUBSTITUTIONS UN^IAIRES. 201 

c*e8t-d.-dire : les coefficients de chaque Equation sout dans le rapport de 

/8, 8 - a, - 7, 
de manifere qu'en supposant que les coefficients a, b, c satisfont k la seule Equation 

(a, 6, cXA S-a, -7) = 0, 
oil 0, i9, 7, 8 sont des quantity quelconques, telles que S + a = 0, on aura 

(a, 5, cXflw? + i9y, yx + Byy = -(QB- fiy)(a, h,c){x, y)». 

Ce n'est \k qu'un cas particulier de T^uation identique 

(a, 6, cXcuc + ^y, yic + Syy + (aS-fiy).{a, b, c){x, y)» = 
(S + a).(aa + 67, 6(S + a), 6/9 + c8) (a?, y)« +08, S-a, -7Xa, 6, c).(/8, S-a, -7Xy, -x)\ 

II &ut remarquer qu'en supposant toujours T^quation 

(a, 6, cX)8, S-a, -7) = 0, 

la fonction quadratique (a, 6, oX^> y)^ ^ supposant qu'elle se riduise d, un carri, est 
comme auparavant P* ou Q*, c'est-il-dire le carrd de Tune des fonctions lin&ires. 
En effet: en ^rivant (a, 6, c)(Xy yY =^ (Ix + myY, T^uation entre i, m serait ^videm- 
ment (/8, S — a, — 7X^» m)" = 0, de mani^re que I, m auraient les m^mes valeurs qu'au- 
paravant. J'ajoute que tout ce qui pr^Me par rapport k T^quation 

(a. by c){(xx + /9y, 7a? + Byy = p (a, 5, c^x, yY 

fait voir qu'^ moins que la fonction quadratique ne soit un carrel, on aura toujours 
p = ± (aS — ^97) ; ce qu'on savait d4jk d^s le commencement, et ce qui pent Stre ddmontrd 
comme a Toidinaire, en ^galant les discriminants (etc — 6") (aS — /97)" et (ac — 6")p' des 
deux cdt^ Je fais enfin p = 1, ce qui donne Tdquation 

(a, 6, cXaa? + fiy, 7a? + 8y)« = (a, 6, c^x, yY, 

et (en faisant abstraction du cas oil la fonction quadratique est un carr^) je tire de 
ce qui prdcMe les r&ultats connus, savoir, que Ton a: 

1. Pour la transformation propre: 

aS-)87 = l, 

a:26:c = 7:8 — a:— /8. 

2. Pour la transformation impropre: 

a/8 + 6(S-a)-C7 = 0. 

Je crois que cette discussion a iti utile pour completer la th^orie alg^brique de la 
forme binaire (a, 5, c){x, y)". 



c. n. 



26 



202 



[137 



137. 



RECHERCHES ULTERIEURES SUR LES DETERMINANTS 

GAUCHES. 



[From the Journal filr die reine und angewandte Mathematik (Crelle), torn. L. (1855), 
pp. 299 — 313: Continuation of the Memoir t. XXXIL (1846) and t xxxviii. (1849); 
5S and 69.] 

J'ai d6jk donn^ une formule pour le d^veloppement d'un cUterminant gauche. 
En prenant, pour fixer les id^es, un cas particulier, soit 



12345 12345 = 



11, 12, 13, 14, 15 

21, 22, 23, 24, 25 

31, 32, 33, 34, 35 

41, 42, 43. 44, 45 

51, 52, 53, 54, 55 

(oii 12 = — 21, &c., tandis que les quantity 11, 22, Sec ne s'^vanoiiiasent pas), 
formule peut dtre ^crite comme suit: 



Cette 



12345 12346= 11 . 


22 . 33 . 


44 . 55 


+ 11 


. 22 . 33 . < 


:45)« 


+ 11 


. 22 . 44 . ( 


[35)» 


+ 11 . 


, 22 . 55 . ( 


[34)' 


+ 11 


, oS . 44 . 1 


[25y 


+ 11 


. 33 . 55 . ( 


[24)' 


+ 11 , 


. 44 . 55 . ( 


[isy 


+ 22 . 


33 . 44 . ( 


;i5)« 


+ 22 , 


, 33 . 55 . < 


[uy 


+ 22 


. 44 . 55 . 1 


(13)* 


+ 33 , 


. 44 . 55 . ( 


[uy 


+ 11 


. (2345)* 




+ 22 


. (1346)' 




+ 33 


. (1245)« 




+ 44 , 


, (1235)« 




+ 55 . 


, (1234)'. 





137] 



RECHEBCHBS ULTJ&RIEURES BUR LES DJ^ERMINANTS QAUCHES. 203 



Lies expressions 12, 1234, <&c k droite sont ici des Pfaffians. On a 

12 = 12, 
1234 = 12.34 + 13.42 + 14.23 

et en Aaivant encore un terme, pour mieux pr^enter la loi: 

123456= 12.34.56 + 13.45.62 + 14.56.23 + 15.62.34 + 16.23.45 
+12.35.64 + 13.46.25 + 14.52.36 + 15.63.42 + 16.24.53 
+ 12.36.45 + 13.42.56 + 14.53.62 + 15.64.23 + 16.25.34. 

J'ai trouvd r^mment une formule analogue pour le ddveloppement d'un dAer- 
fninant gauche bord^, tel que 



cette formule est: 



al234 /31234 = 


; «/8, 


«1. 


a2, 


a3. 


a4 




lA 


11, 


12. 


13. 


14 




2/8, 


21, 


22, 


23. 


24 




3A 


31, 


82, 


33, 


34 




4/8, 


41, 
. 11 


42, 
. 22 


43, 
. 33 


44 


al234 /31234 


= «/9 


. 4 



+ ay8 . 12 . 12 . 33 . 44 

+ a/3 . 13 . 13 . 22 . 44 

+ 0/8 . 14 . 14 . 22 . 33 

+ 0/8 . 23 . 23 . 11 . 44 

+ 0/8 . 24 . 24 . 11 . 33 

+ 0/8 . 34 . 34 . 11 . 22 
+ 0/8 . 1234 . 1234 

+ ol . /81 . 22 . 33 .44 

+ o2 . /82 . 11 . 33 . 44 

+ o3 . /83 . 11 . 22 . 44 

+ 04 . /84 . 11 . 22 . 33 
+ ol23 . /8123 . 44 
+ ol24 . /8124 . 33 
+ ol34 . /3134 . 22 
+ o234 . |8234 . 11. 



26—2 



204 



RECHEBCHES ULT^IRIEURES SUB LES D^TTERMINANTS QAUCHES. 



[137 



II est k peine n^essaire de remarqaer que dans les Pfaffians k droite, oil entrent 
des symboles tels que la, fil, &c, qui ne se trouvent pas dans le determinant dont il 
s'agit, il faut Airire la = — al, /91 = — 1/3, &c. Le symbole /8a ne se trouve ni dans le 
determinant, ni au cdtd droit. Cependant il convient de poser /9a = — a/9 ; car cela extant, 
il serait permis de transformer les P/affians, en ^rivant par exemple a/912 = — /3al2. 

Je remarque en passant que, si avant de poser T^uation /9a s — a/9, on suppose que 
les deux s}rmboles a, fi deviennent identiques (si par exemple on ^rit a = )3 = 5), on 
aurait par exemple 

a/8.12 = a/9.12 + al.2/9 + a2.i91 = 65.12 + 61.25 + 62.51 = 55.12, &c., 

et on retrouverait ainsi la formule pour le ddveloppement de 12346 | 12345. 

La nouvelle formule pent Stre appliqu^e imm^diatement au d^veloppement des 
determinants mineura. En effet, en se servant de la notation des matrices, on aura 



11, 12, 13 
21, 22, 23 
31, 32, 33 



— 1 



123 123 



+ 23 23, - 13 23, - 12 32 



- 23 I 13, + 13 I 13, - 21 I 31 
-3'2Tl2, -STTll, +12 I 12 



11. 
21. 
31. 
41, 



12. 
22, 
32. 

42, 



13, 
23, 
33, 
43, 



14 
24 
34 

44 



— 1 



1234 1234 



+ 234 I 234, - 134 | 234, - 124 | 324, 

- 234 I 134, + 134 | 134, - 214 | 314, 

- 324 I 124, - 314 | 214, + 214 | 214, 
-423 I 123, - 413 | 213, -412 | 312, +123fT23 



-123 1 


423 


-213 


413 


-312 1 


412 



et ainsi de suite. On suppose toujours que chaque terme de la matrice k droite soit 
divis^ par le ddnominateur commun. On voit que les determinants mineurs qui cor- 
respondent k des termes tels que aa, sont des determinants gauches ordinaires, avec le 
signe + , tandis que les determinants mineurs qui correspondent k des termes tels que 

a/9, sont des determinants gauches hordds tels que /9... | a..., avec le signe — . 

Pour donner des exemples de la verification de ces formules, je remarque que Ton 
doit avoir 



123 123= 11 . 23 23 



- 12 . 23 I 13 



-13 . 32 I 12: 
equation qui pent aussi Stre ecrite sous la forme 



123 123= 11 . 23 23 



+ 21 . 23 13 



+ 31 . 32 I 12 



137] BECHEBCHES ULT^RIEURES SUB LES DJ^'ERMmANTS GAUCHES. 



205 



En effet, en d^veloppant les deux cdt^, on obtient: 

11.22.33 + 11.(23)* + 22.(13)' + 33.(12)» = 11.(22.33 + (23)*) 

+ 21.(21.33 + 23.13t) 
+ 31 . (31 . 22 + 32 . 12t). 

On Toit que les deux termes marqu^ par un -f* se d^truisent et que I'^uation est 
identiqtie. On doit avoir de mSme, 



1234 I 1234 = 11 . 234 



- 12 . 234 



- 13 . 324 



- 14 . 423 



ou, ce qui est la mSme chose : 



1234 1234 » 11 . 234 



+ 21 . 234 



+ 31 . 324 



+ 41 . 423 



234 



134 



124 



123, 



234 



134 



124 



123; 



c'est-a-dire, en d^veloppant des deux cdt^s: 



11.22.33.44 + 11.22.(34)" + 11.33.(24)' + 11.44.(23)' 

+ 22.33.(14)' + 22.44.(13)» + 33. 44. (12)' + (1234)' = 

11 [22 . 33 . 44 + 22 (34)* + 33 (42)' + 44 (23)'] 
+ 21 [21 . 33 . 44 + 2134 . 34» + 23 . 13 . 44t + 24 . 14 . 33t] 
+ 31 [31 . 22 . 44 + 3124 . 24* + 32 . 12 . 44t + 34 . 14 . 22t] 
+ 41 [41 . 22 . 33 + 4123 . 23* + 42 . 12 . 33t + 43 . 13 . 22t]. 

Cette expression est en effet identique, comme on le voit en observant que les 
^^^ termes marqu^ par un "f* se d^truisent deux k deux, et que les trois termes 
ntxkjrqu^ par un (*) sont ensemble ^uivalents k (1234)*. 

Je remarque que le nombre des termes du d^veloppement du determinant gauche 
^^ toujours une puiasance de 2, et que de plus, ce nombre se r^uit k la moiti^, 
6^ r^uisant k z6io un terme quelconque act. Mais outre cela, le determinant prend 
^lu cette supposition la forme de determinant d'un ordre inferieur de I'unite. Je 

<^iksidke par exemple le determinant gauche 123 | 123. En y faisant 33 = et en 
Af^centoant, pour y mettre plus de clarte, tous les symboles, on trouve 



123 I 123' = 11' . (23')' + 22' . (13')'. 



206 



RECHKRCHES ULT^RIEUBES SUB LES D^ERMHTANTB GAUCHBS. 



[137 



De la, en ^crivant 



11 = 13'. 11', 12 = 11'. 23', 
22 = 18' . 22', 



on obtient 



12 I 12 = 11 .22 + (12)' 

= 11'.{22'.(13')' + ll'.(23')»). 



12 I 12 = 11' . 123 I 123'. 



c'est-ti-dire 



On a de mSme 

1234 I 1234' = 11' . 22' . (34')» + 11' . 83' . (24')« + 22'33' (14')» + (1234')' 

et de 1^, en ^crivant 

11 = 14'. 11', 12 = 11.24', 23 = 1234', 

22 = 14'. 22', 13 = 11'. 34', 

33 = 14' . 33', 

on obtient 



123 I 123 = 11 . 22 . 83 + 11 . (28)' + 22 . (31)' + 33 (Uy 

= 11' . 14' {22' . 83' . (Uy + (1234')' + 11' . 22' . (34')' + 11' . 33' 



(24')'1, 



c'est-i-dire, 



De mdme 



123 I 123 = 11' . 14' . 1234 I 1234'. 



12345 I 12345' = 11' . 22' . 33' . (45')' 

+ 11' . 22' . 44' . (SSy 

+ 11' . 33' . 44' . (25')' 

+ 22' . 33' . 44' . (15')» 

+ 11' . (2346')« 

+ 22' . (1345')* 

+ 33' . (1245')' 

+ 44' . (lissy. 

Or il est permis d'^rire 

11 = 15'. 11', 12 = 11'. 25', 23 = 1235', 1234 = 2345'. 11'. 15', 
22 = 15'. 22', 13 = 11'. 35', 24 = 1245', 
33 = 15'. 33', 14 = 11'. 45', 34 = 1345', 
44 = 15'. 44'. 

En effet, les quantity k gauche ne sent li^ entre elles que par la seule ^na- 
tion 1234 = 12.34 + 13.42 + 14.23 qui est satisfiodte identiquement par les valeurs i^ 
substituer pour les quantity qui y entrent. Cela dtant, on trouve d'abord: 

1234 I 1234 = 11' (16')' 12345 | 12345'. 



137] 



RBCHBBCHSS ULT^RIEUBES SUR LES DJ^TEBHINANTS OAUCHES. 



207 



Je prends encore on example. On a 



123456 123456'= 11' . 22' . 33* . 44' . 


(56')» 




+ 11' . 22' . 33' . 55' . 


(46')' 




+ 11' . 22' . 44' . 55' . 


(36')' 




+ 11' . 33' . 44' . 55' . 


(2&y 




+ 22' . 33' . 44' . 55' . 


(16')' 




+ 11' . 22' . (U56y 






+ 11' . 33' . (2456')' 






+ 11' . 44' . (2356')» 






+ 11' . 55' . (2346')' 






+ 22' . 33' . (1456')« 






+ 22' . 44' . (1356')' 






+ 22' . 55' . (1346')" 






+ 33' . 44' . (1256')' 






+ 33' . 55' . (1246')* 






+ 44' . 65' . (1236')' 






+ (123456')». 




mis d'^Tire: 






11 = 16'. 11', 


12 = 11'. 26', 23 = 1236', 34 


= 1346'. 45 


22 = 16' . 22', 


13 = 11'. 36', 24 = 1246', 35 


= 1356', 


33 = 16'. 33', 


14 = 11'. 46', 25 = 1256', 




44 = 16'. 44', 


15 = 11'. 66', 




55 = 16'. 55', 






1234 = 2346'. 11 


'.16', 2345 = 128456'. 16', 




1235 = 2366'. ir 


.16', 




1245 = 2456'. 11 


' . 16'. 




1345 = 3456'. 11 


'.16; 





= 1456', 



car les valeun des quantity k gauche satisfont identiqaement aux ^uations qui ont lieu 
entre ces mSmes quantity Par exemple I'^uation 1234 = 12.34 + 13.42 + 14.23 
devient 2346'. 16'= 26'. 1346' + 63'. 1246' + 46'. 1236'. 



Or I'expreasion k droite devient, en d^veloppant: 

26' (13'. 46' + 14'. 68' + 16'. 34') 
+ 63' (12' . 46' + 14' . 62' + 16' . 24') 
+ 46' (12'. 36' + 13'. 62' + 16'. 28'), 



208 



RECHERCHES ULT^RIEUBES SUB LES D^ERMINANTS OAUCHES. 



[137 



et les termes qui contiennent le facteur 16', donnent ensemble 16'. 2346', les autres 
terraes se d^truisent deux k deux. On obtient enfin, en effectuant la substitution: 



12345 I 12345 » 11' . {IQ'y 123456 | 123456' ; 



et ainsi de suite. 



Je fisus les mdmes substitutions dans les matrices inverses, en supprimant cependant 
la derni^re ligne et la demi^re colonne de chaque matrice. On trouve ainsi, en y 
ajoutant les Equations ci-dessus trouv6es par rapport aux determinants: 



13' . 123 I 123' 



11' . 123 I 123' = 12 I 12, 



+ 23 23', - 13 I 23' 



- 23 I 13', + 13 I 13' 



12 12 



-2 2 + 



12 I 12 
11 



+2H, 



. -1|2 



+ lll 



11' . 14' . 1234 I 1234' = 123 | 123, 



14' . 1234 I 1234' 



+ 234 I 234, - 134 | 234, - 124 | 324 
-2347~i24, + 134 | 134, - 214 | 314 
- 324 I 124, - 314 | 214, + 124 | 124 



123 I 123 



- 23 I 23 + 



123 I 123 



11 



- 13 I 23, - 12 I 32 



+ 23 I 13, 

+ 32T12, 



+ 13 I 13, - 21 I 31 
- 31 I 21, + 12 I 12 



11' . (Wy . 12345 I 12345 = 1234 | 1234, 



15' . 12345 I 12345' 



+ 2345 I 2345', - 1345 | 2345', - 1245 | 3245', - 1235 | 4235' 

- 2345 I 1345', +1346 | 1345', - 2145 | 3145', - 2135 | 4135' 

- 3245 I 1245', - 3145 | 2145', +1245 | 1245', - 3125 | 4125' 

- 4235 I 1235', -4135 | 2135', - 4125 | 3125', + 1235 | 1235' 



1234 1234 



- 234 I 234 + 



1234 1234 



11 



+ 234 I 134, 
+ 324T124, 



+ 423 I 123, 



et ainsi de suite. 



- 134 I 234, - 124 | 324, - 123 | 423 



+ 134 I 134, - 214 I 314, - 213 | 413 

- 314 I 214, + 124 I 124, - 312 | 423 

- 413 I 213, - 412 I 312, + 123 I 123 



137] 



BECHERCHES ULTl^IEUBES SUR LBS D^TTEBMINAITTS OAUCHES. 



209 



II est bon de changer un peu la forme de ces ^uationa On en d^uit sans 



peine 



13' . 123 I 123' 



2 . 23 I 23' - 123 123* 



11 



, - 2 . 13 I 23' 



- 2 . 23 I 13', 



+ 2.13 I 13'- 



, 123 123' 



22 



12 12 



-2.2| 2+ 



12 I 12 



11 



, -2.1|2 



+ 2.2 1, 



+ 2.1 1- 



12 12 



22 



^'.123411234' 



2 . 234 234'- 



, 1234 1234' 



11' 



, -2.134 I 234', 



-2.124 324' 



-2 . 234 I 134', 



2 . 134 134'- 



, 1234 1234' 



22' 



, -2 . 214 I 314' 



-2 . 324 I 124', 



-2.814 I 214', 



..mnsi-'.^*^ 



123 123 



2 . 23 I 23- 



123 123 



11 



- 2 . 13 23, 



+ 2.23 I 13, 



+ 2. 13 I 13- 



123 123 



22 



+ 2.32 ! 12, 



- 2 . 81 21, 











-2 


,12 


32 


-2 


.21 


31 

• 




123 1 123 


+ 2. 


12 


12- 


33 



et ainsi de suite. 

Les formules que je viens de presenter, peuvent 6tre appliqu^ aussitdt k la 
solution de la question suivante: "Trouver Xi, x,, x„ &c., fonctions lin^ires de Xi, 
Xi, Xj, &c. telles que 

11 Xi« + 22 x,»+ 33 x," + &c. = 11 a?i" + 22 a:,» + 33 a?,« + &c./' 

c*est-it-dire : transformer une fonction quadratique llari' + 22ajj' + 33a:b' + &c. en elle- 
mSme par des substitutions lin^aires. II suffira d'^rire la solution pour le cas de 
trois ind^termin^: on satisfait identiquement k I'^uation 

11 Xi« + 22 x,» + 33 x,« = 11 a?i» + 22 a?,* + 33 a?,» 

en ^rivant 

1 



(Xj, Xj, Xj) — 



123 I 123 



+2.23 I 23 



123 123 



11 



, -2.13 I 23, 



-2.12 I 32 



(ll«i, 22«,. 33a^). 



-2.23 I 13. 
I-2.327T2, 



+ 2.13 I 13- 



123 123 



22 



, -2.21 131 



-2.31 I 21, 



+2.12 12- 



123 I 123 
33 



C. II. 



27 



210 



RECHEBCHE8 ULTI^EUIIES SUB LES Dl^TEBMINANTS GAUCHES. 



[137 



Voil^ la transformation propre. On en tire la transformation impropre de ll^i' + 22d!:,'' 
en dle-mSme en ^rivant 33 = 0; car, cela pos^, les valeurs de Xi, x, ne contiennent 
pas Xg, et Ton n'a plus besoin de la valeur de x,. On obtient ainsi la solution 
suivante; savoir, on satis&it identiquement h T^uation 



en ^rivant 



(x,, Xj) = 



123 I 123' 



ir Xi» + 22^ x,» = ir a?i« + 22^ a?,« 



2.23 |23'-^?^jL^', -2. 13 I 23' 



(ira?», 22-^:,), 



- 2 . 23 I 13', 



2 . 13 I 13 



, 123 I 123^ 
22 



ce qui est une transformation impropre. Mais en y fiusant la substitution 11 =13'. 11', 
22 = 13'. 22', 12 = 11'. 23', on r^uit la solution k celle-ci, savoir on satis&it identique- 
ment k r^uation llxi" + 22x,"= lla?i» + 2aci" en ^crivant 



(xi, ^fd- 12 I 12 



1 O TO . 

-2.21 2 + i^^, -2.112 



11 



+ 2 . 2|1, 



+2.11- 



12 I 12 
22 



(lla^i, 2ac), 



ce qui est encore une transformation impropre, qui correspond de plus pr^ a la 
formule pour la transformation propre; la seule diffi^rence est que les signes des 
termes de la premiere colonne de la matrice en sent chang^ 

En introduisant des lettres simples a, 6, &c & la place des symboles 11, 22, &c., 
je consid&re d'abord la transformation propre 



ax* + 6y* = da^ + 6y*. 



Ici, en 6criyant 



11. 


12 


^ 


a, p 


21. 


22 




-V, h 



la formule donne 



(X, y) = 



ab + ^ 



ab-i^, - 2i* 
2va , ab'-i^ 



(«. y)- 



La transformation impropre 



ax* + 6y* = ewB* + 6y* 



s'obtient au- moyen de la formule donnee plus bas pour la transformation propre de 
la fonction aa^ + bi/* + c^ en eUe-mSme, En y &srivant c = 0, on obtient 



(X, y) = 



aX« + 6^» 



2\fia , — aX* + bfi* 



(«i y)- 



137] 



RBCHEBCHE8 ULT^RIEUIIES SUB LES D]fer£ItMINANTS GAUCHES. 



211 



J'ai d4jk £Edt voir de quelle mani&re cette formule se rattache k la formule pour 
la transformation propre; la difif<^rence entre les formes de ces transformations dans ee 
cas simple est assez firappante« 

Pour obtenir la transformation propre 



ax' + 6y' + cz* = cur* + 6y' + C2^, 



j'&ris 



11. 


12, 


13 


= 


21. 


22. 


23 




31. 


32. 


33 





v, 6, 



X 
c 



eette formule donne 



(x, y, z) = 



abc + aK*-^ bfA* + ci^ 



a6c + aX*-6|A*-ci^, 2(\fjh-cv)b , 2(vX + 6aa)c 

2(X|A+cv)a , abc — aX^ + bfjL^^cp', 2(jiv — a\)c 
2 (vX — bfi) a , 2 (/LM/ + aX) 6 , a6c — aX* — 6/a' — ci^ 



La transformation impropre 0n eUe-mSme 

ax' + 6y' + cz' = oa?" + 6y* + c^' 



(^. y. z)' 



^tre tir^ de la transformation propre en die-mime de la fonction donn^e ei-apr^s 
-f 6y' + C2^ + rft(;*; en y ^crivant d = 0, on obtient 



^^* ^' ^^ 6cp«+ca<r» + a6T« + <^» 



bcp^ ■{■ cao^ •\' abi* - 4^, - 2bT(l> - 2bcp(r , 2ca<f>-2bcTp 

2ar4> — 2ac/!>cr , ic/^* — caa* + air" — 0*, — 2cp<l> — 2ca<rT 

2aa(l> — 2abpT , 2bp4>-2abaT , bcp* -h caa* ^ abi* — (f)^ 



(«^» y, '^). 



Pour verifier que cette expression n'est en effet autre chose que la formule pour 
^ transformation propre, en y changeant les signes de tous les termes, j'^ris dans la 
^^Hxiule pour la transformation propre, as=& = c = a>. On a ainsi pour la transformation 
P^pre 

x' + y* + z' = a;« + y* + -8^, 

^ ^nation 



(t. 






cD«+X*-/A*-i'', 2X|A-2vai , 2i^X+2^cD 
2X^ + 21^0 , ®»-X* + /A«-i;*, 2^1^-2X01 
2i/X-2/*® , 2/Av + 2Xa> , ««-X« + aa"-»^ 



(^, y, '^), 



27—2 



212 



RECHEBCHBS ULT^BIEnitES SUB LES Dl^EBMINANTS GAUOHES. 



[137 



et en ^rivant dans la formule pour la transformation impropre, a=& = c = l, d^^O, et 
\, fi, V, — m au lieu de p, cr, r, <!>, on obtient pour la transfonnation impropre 



r^uation 



x« + y«+2? = a;« + y*+-8*, 



a)*-X* + ^« + i^*, -2\|A+2w , -2i^X-2/*« 
-2X/i-2i^a) , -a)* + X*-A** + i^, 2/iav-2X» 

-2ifX + 2/u» , -2/iAv-2\i» , -©« + X"-At*-i^ 

Pour obtenir la transformation propre 

ax' - 6y' + cz' + rfw' = flur* + fty' + ci* + du^, 

11, 12, 13, 14 

21, 22, 23, 24 

31, 32, 33, 34 

41, 42, 43, 44 

oela donne d'abord, en mettant pour abr^ger, 
la valeur du determinant 



(^» y» ^)' 



j'&ris 



a, 


Vf 


-M, 


P 


V, 


6, 


X, 


<T 


M, 


-X, 


c, 


T 


P> 


-<r. 


-T, 


d 



— 1 



1234 I 1234==aJcd + 6c/5« + ca<r* + a67* + acrx« + M/i» + c(ii;« + ^ 
(ce que je repr^sente par &). 

J'ajoute aussi la valeur de la mairicA inverse 

11, 12, 13, 14 

21, 22, 23, 24 

31, 32, 33, 34 

41, 42, 43, 44 



ocd + 6t* + c/E)* +dfi*, 
ddK-^rp^ + dfUf^aoTt 
aca + fjLtf} -^ cvp +aXT, 

bdfi H- o-if} + dXjf — bpT , 

— odK ^pif) + dfjtv — aoT, 

abd + aa* + bf^ +dv^ , 

dbr + 1^^ + bfip — aXo", 



savoir: 


1 


bed +bT* +ca* +(iX«, 


k 


cdp + T^ + dXft — cpa. 




— Mft — cr^ + (iXi^ — bpr, 




bcp +X^ + ci'cr ^bfir, 



— 6cp — X^ + cva — 6/iT 
•- aca — fuf) — cpp 4-aXT 

— air — 1^0 + 6/Lif) — a\a 
abc +aX'+6/i« + ci/» 



137] 



BBCHERCHES ULT^RIEURES SUR LES Dl^RMINANTS GAUCHES. 



213 



On a pour la transformation, I'^uation (x, y, z, w) = 



1 
k 



abed — 6cp' + ccur* + abi^-h adX? 

2a (cdv + T0 + dXfjk — cp<r) , 

2a (— bdfi — cr^ + dKv — bpr), 
2a (bcp + X^ + cvtr — bfir) , 



26 (— cdv — T^ + dKfi — cpcr) , 
a6c(i + bcp* — ccur* + abr* — ad\" 

26 (adk + p^ + rf^v — a<rr) , 
26 (oca + fjul> — cvp + aXr) , 



2c (6d/i + cr^ + (iXv — 6pT) , 

2c (— adX — p<l> + dfjkv — cutt) , 

a6cd + bcp^ + ccur" — a6T' — odX* 
- 6d^« + cdi/* - 0« 

2c (a6T + 1^ + bfjLp — aXo*) , 



2d (— 6cp — X0 4- ci'cr — bfir) 
2d (— accr — |i^ — ci'p + aXr) 

2d (— abr — i^^ + 6/ip — aXo") 
abed — 6c/!)' — caa* — abr* 4- adX* 



(or, y, J?, m;). 



Je suppose que Ton ait a = 6 = c = d = «, et j'&jris -^ = , c'est-i-dire 

^ = — ^- -^*^ — ^ ou \p + /xo- + in- + '^a) = 0. En fiedsant cette substitution, on trouve 
d'abord k — fu^B, oh 

iJ = X« + ;i« + l;« + ^ + p« + <T« + T»+a)«, 

et puis pour la transformation propre 

x« 4- y « 4- z« + w' = a* + y ' + -8^ + W, 
TAjuation (x, y, z, w) = 

-p« + <r* + T« + tt)* + X"-^"-I^-'^, 

2a>v - 2t'^ + 2X/i - 2/!)cr, 

- 2®/i + 2<r^ + 2Xi/ -2/yr, 

2a>p - 2X^ + 2i^cr - 2/iat, 



1 



- 2o>v + 2t^ + 2X/i - 2pcr, 

2a)X - 2p'^ + 2^1^ - 2<rr, 
2a>(r - 2/i^ - 2i;/o + 2Xt, 



2mp. - 2<r^ + 2Xv —2pT 
2oiX + 2p^ + 2/u/- 2<rr 
p« + <r» - T» + «»- X«- AA« + i^ 
2toT- 2i^+ 2aap - 2Xcr 



-^, - 



2aip + 2X^ + 2va - 2/iT 
2a)cr + 2^^ - 2pp + 2Xt 
2ftrr + 2ir^ + 2/i/) - 2X<t 



-^' 



(a:, y, ^, w). 



On peut changer la forme de cette expression, en y &rivant 



X = i(«-aO. /t = i(/9-i8'). i' = i(7-7'). V^ = i(S-8'). 



214 



BECHEBCHES ULTl^RIEUBES SUR LES Dl^TEBMINANTS GAUCHES. 



[137 



cela donne 






de mani^re qu'en ^crivant 



on obtient 



R = \{M-\-M')^.J{MM'), 



et la formule pour la transformation devient 



(x, y, z, w) = 



V {MM') 



aa' + i8/8' + 77' +S8', 
a/8'-i8a'+78'+V. 



cu^-fifi'+yy'+iS', 

aS' - ^' - yfi' - Sc^ . 

-»y'-y8S' + 7««'-Si8', 



»y' +/SS' 

oS' —fiy' 



yd - 1?, 

y^ + Scl. 
77' + SS*, 
7S' - S7'. 



oy + ies* + 70' + s^ 

afi'-ficl+y^ + Sy' 
ad -^ff-yy'+BS' 



(x. y, z, w). 



la 



Oil voit done que mdme sans supposer T^uation M = M\ cette formule donne 
transformation propre 

x« 4- y« 4. z* + w* = a;* + y* + ^ + ?*;". 

Cette solution est k peu pr^ de la m6me forme que la solution impropre donn^ 
par Euler dans son m^moire "Problema algebraicum ob affectiones prorsus singulares 
memorabile " Nov. Comm. Petrop., t. xv. 1770, p. 75, et Coram. Arith. collects, [4to. 
Petrop. 1849], t. i. p. 427. Je remarque aussi que cette ' m6me solution pent 6tre 
d^uite de la th^orie des Quaternions, En effet, t, j, k ^tant des quantity imaginaires 
telles que i*=j' = A;* = — 1, jfc = — ^' = t, ki — — ik^jy %j^''ji=sk, on obtient, en effectuant 
la multiplication: 

X, y, z, w ayant les mdmes valeurs que dans la demi^re formule de transformation. 

En changeant les signes des termes de la quatri&me colonne, on en tire pour la 
transformation impropre 

x' + y' + z' + w* = a:' + y* + -»• + w*. 



137] 



RECHERCHES ULTiRIEURES SUR LES D^^TERMINANTS OAUCHES. 



215 



la fonnule suivante plus sym^trique : 



(X, y, z, w)-^^^j^^ 



-oa' +/9/8'+ry' + S8', -ciff 
-ev9'-/8a'+78'-&y', aa' 

-»y'-y88'-7a'+S/8', US' 



/9a' - 78' + 87' , 



aiSf -fir/ 

ettt'+ff^ 
aff-fia' 



To'-S/y, 

77'+SS'. 
7«' - V, 



ay - ^87' + 7)8' - So' 
a7' -ffS' -yc^ -SIS' 
afi'+fic^ -yB' -By' 
a«'+y8/8' + 77'-88' 



(a;, y, «, w). 



Ces formules pour la transformation, tant propre qu'impropre, de la fonction 
^ + y* + '8* + W en eUe-mSme, sont utiles dans la th^rie des polygenes inscrits dans 
line sur£Eu» du second ordre. 



216 



[138 



138. 



RECHERCHES SUR LES MATRICES DONT LES TERMES SONT 
DES FONCTIONS LINEAIRES D'UNE SEULE INDfiTERMINEE. 

[From the Jounial fiir die reine und angewandte Mathematik (Crelle), torn. L. 

(1855). pp. 313—317.] 



Je pose la matrice 



A y IJ I G , ... 

A I IJ , G , ... 



dont les termes {n* en nombre) sont des fonctions lin^aires d'une quantity «, et je 
considfere le determinant form^ avec cette matrice, et les determinants mineurs fonn& 
en supprimant un nombre quelconque des lignes et un nombre ^gal de colonnes de 
la matrice. En supprimant une sevle ligne et une aeule colonne, on obtient les 
premiers mineurs; en supprimant deux lignes et deux colonnes, on obtient les seconds 
mineurs; et ainsi de suite. Cela ^tant, je suppose que la quantity s a ^t^ trouvde 
en egalant k z^ro le determinant form^ avec la matrice donn^e ; ce determinant sera 
une fonction de s du n-i^me degre qui g^neralement ne contiendra pas de facteurs 
multiples. On voit done qu'un facteur simple du determinant ne pent pas entrer comme 
facteur dans les premiers mineurs (c'est-k-dire dans tous les premiers mineurs) ; mais 
en supposant que le determinant ait des facteurs multiples, un fisu^teur multiple du 
determinant pent entrer comme facteur (simple ou multiple) dans les premiers mineurs, 
ou dans les mineurs d*un ordre plus eieve. II importe de trouver le degre selon lequel 
un facteur multiple du determinant pent entrer comme facteur des premiers mineurs, 
ou des mineurs d'un ordre quelconque donne. 

Cela se fait trfes facilement au moyen d'une propriete generale des determinants; 
si les mineurs du (r4-l)ifeme ordre contiennent le facteur («— a)* (c'est-it-dire, si tous 



138] BECHEBCHES SUB LES MATBICES &C. 217 

les mineurs de cet ordre contiennent le facteur (« — a)*, mais non pas tous les fistcteurs 
(» — a)*"*"*) ; et si de m^me les mineurs du r-ifeme ordre contiennent le fiskcteur (« — a)^ ; 
alors les mineurs du (r — l)i&me ordre contiendront au moina le £Etcteur (« — a)^^"*. 
^utrement dit: les mineurs du (r — l)ifeme ordre contiendront le facteur (« — a)^ od 
'y > 2)8 — a, ou, ce qui est la mfime chose, a — 2/8 + 7 <t ; c'est-k-dire : en formant 
la suite des indices des puissances selon lesquelles le facteur {s — a) entre dans les 
Tuineurs premiers, seconds, &c. (il va sans dire que cette suite sera une suite dScroissante), 
lee di£f(^rences secondes seront positives [c'est-k-dire non nigativ€s\ Je repr^sente par 
^<> /8, 7, ... la suite dont il s'agit; je suppose, pour fixer les iddes, que h soit le 
ciemier terme qui ne s'^vanouisse pas, et j'^cris 

a, ^, 7, S, 0, 0, . . . 

a-A )8-7, 7-S, S, 0,... 

a-2;9 + 7. i8-27 + S, 7-2S, S, 0,...; 

xci, quel que soit le nombre des termes, tous les nombres de la troisi^me ligne seront 
positi&, et en reprdsentant ces nombres par /, /', /'', &c., on obtient: 

a=/+2/' + 3r + 4r+..., 

^= /'+2r+3r+..., 
7= /-+2r+..., 

n y a ici k consid^rer que le nombre a, indice de la puissance selon laquelle le 

facteur {s — a) entre dans le determinant, est donn^ ; il sera done permis de prendre 

jy>ur /, /', /",... des valeurs enti^res et positives quelconques (z^ro y compris) qui 

aatisfont k la premiere ^nation; les autres ^nations donnent alors les valeurs de 

iS, % Sy &c On forme de cette mani&re une table des particularity que pent printer 

un facteur multiple {s — a)* du determinant ; cette table est compos^e des symboles 

^> /3, y,..., et les nombres a,l3,.., de chaque symbole font voir le degr^ selon lequel 

le £acteur (s — a) entre dans les determinants, dans les mineurs premiers, seconds, &c. 

Or il est trfes facile de former, au moyen des tables pour a^l, a = 2, ... as A;, la table 

pour a=sit+l. On a par exemple pour a=l, a = 2, a = 3, a = 4t les tables suivantes: 

Pour a = 1, 1. 

Pour = 2, 2, 21. 

Pour a = 3, 3, 31, 321. 

Pour = 4, 4, 41, 42, 421, 4321. 

De \k on tire la table pour a = 5, savoir : 

Pour a = 6, 6, 61, 62, 621, 631, 6321, 64321. 

£n eifet, le premier terme est 6, et on obtient les autres termes en mettant le nombre 
) devant les symboles des tables pour a^l, 0=2, a = 3, a = 4, en ayant seulement soin 
C. II. 28 



218 RECHGRCHGS SUR LES MATRICES DONT LES TBBHES 30NT [l38 

de supprimo- les symbolee 58, 54-, 541, 542, 5421 pour lesquels le premier terme de 
la Buite des diffdreDces eecondes est nigsAif. On trouve de mSme pour a = 6, la table 
suivante, savoir : 



Pour a = 6 



6, 61, 62, 621, 63, 631, 6321, 642, 6421, 64321, 654321; 



et ainsi de suite. Lea nombrea dea Bymboles pour a = l, 2, 3, 4, 5, 6, 7, 8, &c sont 
1, 2, 3, 5, 7, 11, 15, 22, [30, 42, 56], &c.; ce aont les coefficients des pmBSonces a^. of, a? 
&C. dans le d^veloppement de 

(1-ic)-' (l-a:')-' {l-x'y* (1-3-)-' (l-«^)-' ... &c. 

foQCtions qui se pr^entent, comme on sait, daos la th^rie de la pari^on des nombrea. 

Maintenant, au lieu de coosiddrer un seulfacteur du determinant, J e considfere toils 
les facteurs: par exemple pour n = 4, le determinant pout avoir un fiacteur double (s—df, 
et un autre facteur double (« — 6)* ; il peut de plus arriver que le facteur (« — a) soit fiwteur 
simple des premiers mineurs, mais que le facteur (s ~ h) n'entre paa dans lee premiers 
mineurs. Le eymbole qui correspond au &cteur (s — a) sera 21, et le symbole qui 
coirespond au facteur (B—h) sera 2. En combinaot cee deux symboles, on aura le 



symbole composd 



qui denote que le determinant a deux facteurs doubles de 



la clasee dont il s'agtt. Je forme de ces symboles composes 
n = 2, n = 3, 7i = 4, &c. On a: 



table* pour n = 1 , 



^0' 



1 




2 


. 21 


1 




1 


1 


_ 









. 31 , 321 





Pour 


n = 


4 : 


















1 




2 




21 




3 




321 




2 




21 


1 
1 
1 




1 
1 




1 




1 




1 




2 




2 






^ 



















et ainsi de suite. 



g.g, 



42 . 421 , 4321 [ 



138] DES FONCTIONS LINJ&AIRES d'UNE SEULE INDlfeTERMINilE. 219 

321 
Pour donner encore un exemple du sens de ces symboles, le symbole denote 




cjue le determinant a un facteur (s — a) qui entre comme facteur triple dans le deter- 
minant, comme &eteur double dans les premiers mineurs, et comme facteur simple dans 
les seconds mineurs; Tautre facteur du determinant est un facteur simple (s — b), Les 
xiombres des symboles pour n = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, &c. sont 1, 3, 6, 
14, 27, 58, 111, 223, 424, 817, 1527, &c. ; ces nombres sont les coefficients de oc", a?, a*, 
<&c, dans le developpement de 

( 1 - jr)-^ (1 - aj»)-« (1 -a^)-'(l -ic*)-» (1 - a^y (1 - a:*)"" (1 - a^)-"(l - af)-^ (1 - a;»)-». . . &c. 

cju les indices 1, 2, 3, 5, 7, 11, &c. forment la suite qui se pr^sente dans la theorie 
c^e la partition des nombres, dont j'ai parie plus haut II est tr^s facile de d^montrer 
f^u'il en est ainsi. 

Les r^sultats que je viens de presenter sont en partie dus k M. Sylvester (voyez 
isson m^moire "An enumeration of the contacts of lines and surfaces of the second 
cz^rder," Philosophical Magazine, [voL I. (1851), pp. 18 — 20]). En efFet, M. Sylvester 
ciommence par etendre h. des fonctions d*un nombre quelconque d'indeterminees Tid^e 
^^ometrique des contacts des courbes et des surfaces. £n considerant les deux ^qua- 
't.ions quadratiques [7=0, F=0, il forme le discriminant de la fonction quadratique 
ZJ -\- aV^ et ir cherche dans quel degr^ chaque facteur de ce discriminant pent entrer 
c^romme facteur dans les mineurs premiers, seconds, &c. Le discriminant de M. Sylvester 
<=^st un determinant sym^trique ; mais cela ne change rien k la question, et je n'ai 
fjEiit que reproduire Tanalyse de M. Sylvester, en donnant cependant Talgorithme pour 
la formation des symboles, et de plus la loi pour le nombre des symboles. M. Sylvester 
cJonne pour n = 2, 3, 4, 6, 6, des nombres qui, en ajoutant k chacun le nombre 2, 
pK>ur embrasser deux cas extremes qui ne sont pas comptes, seraient 3, 6, 14, 26, 58. 
X\ se trouve dans le nombre 26 une erreur de calcul ; ce nombre devrait 6tre 27, et 
^n suppliant le premier terme 1, on a la suite trouv^e plus haut, savoir 1, 3, 6, 14; 
27, 58, &c. ; il y a de mSme une erreur de calcul dans les nombres donnas par M. 
Sylvester pour n = 7 et n = 8. 

Mais tout cela s'applique k une autre theorie g^ometrique, savoir k la theorie des 
figures homographes. Pour fixer les id^es, je ne considere que les figures dans le plan. 
Kn supposant que x, y, z soient les coordonnees d*un point, et en prenant pour (x, y, z) 
cles fonctions lin^aires de (ar, y, z) on aura (x, y, z) comme coordonnees d'un point 
honiographe au point (x, y, z). En cherchant les points qui sont homographes chacun 
^ soi-mfime, on est conduit aux Equations x— «a? = 0, y — «y = 0, z — «z = 0. Les 
c^uantit^s k gauche x — sx, y — «y, z— sz sont des fonctions lindaires de x, y, z, ayant 
pour coefficients des fonctions lin^aires de s. On a ainsi une matrice dont les termes 
^ont des fonctions lin^aires de s] la th^orie enti^re se rattache aux propriet^s de 
eette matrice. Pour le cas gdndral de Vhomographie ordinaire, on a le symbole 

28—2 



220 



RECHERCHES SUB LES MATRICES &C. 



[138 



1 
1 
1 



, pour Yhomologie, le symbole 




; les autres symboles 



2 
1 



• IB- 



31 



se 



321 



au cas de Tidentit^ 



rapportent k des cas moins g^draux, et le symbole 

complete des deux figures; y compris ce cas-limite de Tidentit^ complete, il eziste 
pour le plan 6 espfeces d'homographie ; pour Yespace ordinaire il existe 14 especes 
dliomographie. Je reviendrai k cette th^rie k une autre occasion. 



LtmdreSy le 24 Mai 1854. 



139] 



221 



139. 



AN INTRODUCTOKY MEMOIR UPON QUANTICS. 

[From the Philosophical Transactions of the Royal Society of London, vol. cxliv. for the 
year 1864, pp. 244—258. Received April 20,— Read May 4, 1854.] 

1. The term Quantics is used to denote the entire subject of rational and integral 
fimctiens, and of the equations and loci to which these give rise ; the word " quantic " 
is an adjective, meaning of such a degree, but may be used substantively, the noun 
understood being (unless the contrary appear by the context) function; so used the 
word admits of the plural •' quantica" 

The quantities or symbols to which the expression "degree " refers, or (what is the 
same thing) in regard to which a function is considered as a quantic, will be spoken 
of as "facients." A quantic may always be considered as being, in regard to its 
facients, homogeneous, since to render it so, it is only necessary to introduce as a 
facient unity, or some symbol which is to be ultimately replaced by unity; and in the 
cases in which the facients are considered as forming two or more distinct sets, the 
quantic may, in like manner, be considered as homogeneous in regard to each set 
separately. 

2. The expression "an equation," used without explanation, is to be understood as 
meaning the equation obtained by putting any quantic equal to zero. I make no 
absolute distinction between the words "degree" and "order" as applied to an equation 
or system of equations, but I shall in general speak of the order rather than the 
degree. The equations of a system may be independent, or there may exist relations 
of connexion between the different equations of the system; the subject of a system 
of equations so connected together is one of extreme complexity and difficulty. It will 
be sufficient to notice here, that in any system whatever of equations, assuming only 
that the equations are not more than sufficient to determine the ratios of the facients, 
and joining to the system so many linear equations between the facients as will render 
the ratios of the facients determinate, the order of the system is the same thing as 
the order of the equation which determines any one of these ratios; it is clear that 
for a single equation the order so determined is nothing else than the order of the 
equation. 



222 AN INTRODUCTORY MEMOIR UPON QUANTICS. [l39 

3. An equation or system of equations represents, or is represented by a locus. 
This assumes that the facients depend upon quantities x^ yi ••• the coordinates of a 
point in space ; the entire series of points, the coordinates of which satisfy the equation 
or system of equations, constitutes the locus. To avoid complexity, it is proper to take 
the facients themselves as coordinates, or at all events to consider these facients as 
linear functions of the coordinates; this being the case, the order of the locus will be 
the order of the equation, or system of equations. 

4. I have spoken of the coordinates of a point in space, I consider that there is 
an ideal space of any number of dimensions, but of course, in the ordinary acceptation 
of the word, space is of three dimensions; however, the plane (the space of ordinary 
plane geometry) is a space of two dimensions, and we may consider the line as a space 
of one dimension. I do not, it should be observed, say that the only idea which can 
be formed of a space of two dimensions is the plane, or the only idea which can be 
formed of space of one dimension is the line ; this is not the case. To avoid complexity, 
I will take the case of plane geometry rather than geometry of three dimensions; it 
will be unnecessary to speak of space, or of the number of its dimensions, or of the 
plane, since we are only concerned with space of two dimensions, viz. the plane ; I say, 
therefore, simply that x, y, z are the coordinates of a point (strictly speaking, it is the 
ratios of these quantities which are the coordinates, and the quantities x, y, z themselves 
are indeterminates, i.e. they are only determinate to a common factor prhs, so that in 
assuming that the coordinates of a point are a, )8, 7, we mean only that x : y : z^a : 0:y, 
and we never as a result obtain x, y, z = a, /3, 7, but only x : y : z^a : /3 : y; but 
this being once understood, there is no objection to speaking of x, jf, z as coordinates). 
Now the notions of coordinates and of a point are merely relative; we may, if we 
please, consider x : y : z as the parameters of a curve containii^ two variable para- 
meters; such curve becomes of course determinate when we assume x : y : z^a : fi : y, 
and this very curve is nothing else than the point whose coordinates are a, 0, 7, or 
as we may for shortness call it, the point (a, /3, 7). And if the coordinates (x, y, z) are 
connected by an equation, then giving to these coordinates the entire system of values 
which satisfy the equation, the locus of the points corresponding to these values is the 
locus representing or represented by the equation ; this of course fixes the notion of a 
curve of any order, and in particular the notion of a line as the curve of the first 
order. 

The theory includes, as a very particular case, the ordinary theory of reciprocity in 
plane geometry; we have only to say that the word "point" shall mean "line,* and the 
word "line" shall mean "point," and that expressions properly or primarily applicable 
to a point and a line respectively shall be construed to apply to a line and a point 
respectively, and any theorem (assumed of course to be a purely descriptive one) relating 
to points and lines will become a corresponding theorem relating to lines and points; 
and similarly with regard to curves of a higher order, when the ideas of reciprodtv 
applicable to these curves are properly developed. 

5. A quantic of the degrees m, m'... in the sets (a?, y...), {x\ y'...) &a will for the 
most part be represented by a notation such as 

m m' 

(♦$^, y...$a:', y'... )...). 



139] AN INTRODUCTORY MEMOIR UPON QUANTICS. 223 

^where the mark * may be considered as indicative of the absolute generality of the 

qaantic; any such quantic may of course be considered as the sum of a series of 

therms afy^...fl/*y^..., &c. of the proper degrees in the different sets respectively, each 

-term multiplied by a coefHcient ; these coefficients may be mere numerical multiples 

of single letters or elements such as a, b, c,..., or else functions (in general rational 

siod integral functions) of such elements ; this explains the meaning of the expression 

*' the elements of a quantic": in the case where the coefficients are mere numerical 

xnnltiples of the elements, we may in general speak indifferently of the elements, or 

of the coefficient& I have said that the coefficients may be numerical multiples of 

single letters or elements such as a, b, c, ...; by the appropriate numerical coefficient 

of a term afy^...a:'*y^ ..., I mean the coefficient of this term in the expansion of 



m 



(a: + y...) (a/ + y'... )...); 
^nd I represent by the notation 

m m 

(a, b,...^x, y,...$J?', /,...)...)> 
quantic in which each term is multiplied as well by its appropriate numerical coeffi- 



cient as by the literal coefficient or element which belongs to it in the set (a, 6,...) of 
literal coefficients or elements. On the other hand, I represent by the notation 



m m 



(tt, b,...\x, y,...^a?', /,...)...)» 

quantic in which each term is multiplied only by the literal coefficient or element 
"vvhich belongs to it in the set (a, &,...) of literal coefficients or elements. And a like 
distinction applies to the case where the coefficients are functions of the elements 
(a, 6, ...). 

6. I consider now the quantic 

« m' 

^Knd selecting any two facients of the same set, e.g. the facients x, y, I remark that 
t^here is always an operation upon the elements, tantamount as regards the quantic 
t:o the operation ady; viz. if we differentiate with respect to each element, multiply 
V)y proper functions of. the elements and add, we obtain the same result as by differ- 
entiating with 3y and multiplying by x. The simplest example will show this as 
"^rell as a formal proof; for instance, as regards Saa^ -^ bxy -\- 5cy^ (the numerical 
Coefficients are taken haphazard), we have ^a + lOcd^ tantamount to xdy', as regards 
c^ (a: — ay) (^ — i8y ), we have — a(a+/8)3a + a^«4-j8^^ tantamount to ady, and so in any 
cither case. I represent by [ady] the operation upon the elements tantamount to aSy, 
s^nd I write down the series of operations 

'Where x, y are considered as being successively replaced by every permutation of two 
clifferent £Eu;ients of the set (a:, y,...); ^> }/ as successively replaced by every permutation 
of two different facients of the set {x\ y',...)> fti^d so on; this I call an entire system, and 



224 AN INTEODCJCTORY MEMOIB UPON QUANTIC8. [l39 

I say that it is made up of partial systems ooiresponding to the different fiunent sets 
respectively; it is clear from the definition that the quantic is redaoed to zero by 
each of the operations of the entire system. Now, besides the quantic itself, there 
are a variety of other functions which are reduced to zero by each of the operations 
of the entire system; any such function is said to be a oovariant of the quantic, and 
in the particular case in which it contains only the elements, an invariant. (It would 
be allowable to define as a covariant quoad any set or sets^ a function which is reduced 
to zero by each of the operations of the corresponding partial system or systems, but 
this is a point upon which it is not at present necessary to dwell) 

7. The definition of a covariant may however be generalized in two directions: 
we may instead of a single quantic consider two or more quantics; the operations 
{aSy], although represented by means of the same symbols x, y have, as regards the 
different quantics, different meanings, and we may form the sum 2 {^y}> where the 
summation refers to the different quantics: we have only to oondder in place of the 
system before spoken of, the system 

2{a3y} — aj9y, ... ; 2{a:'3y'} —x'd^y ... &c. &c., 

and we obtain the definition of a covariant of two or more quantics. 

Again, we may consider in connexion with each set of £Eu;ients any number of 
new sets, the facients in any one of these new sets corresponding each to each with 
those of the original set; and we may admit these new sets into the covariant. This 
gives rise to a sum 8[a!d^y where the summation refers to the entire series of cor- 
responding sets. We have in place of the system spoken of in the original definition, 
to consider the system 

[ocdy] '-S{aSy\ ... [x'd^] — 8 {afd^\ ... &c. Ac., 

or if we are dealing with two or more quantics, then the system 

2 [xdy] — S{ady), ... ; 2 {^'Sy'} '-S(x'd^), ... &c. &c., 
and we obtain the generalized definition of a covariant. 

8. A covariant has been defined simply as a function reduced to zero by each oi 
the operations of the entire system. But in dealing with given quantics, we may 
without loss of generality consider the covariant as a function of the like form with 
the quantic, Le. as being a rational and integral function homogeneous in regard to 
the different sets separately, and as being also a rational and integral function of the 
elements. In particular in the case where the coefficients are mere numerical multi- 
ples of the elements, the covariant is to be considered as a rational and integral 
function homogeneous in regard to the different sets separately, and also homogeneous 
in regard to the coefficients or elementa And the term "covariant" includes, as already 
remarked, "invariant." 

It is proper to remark, that if the same quantic be represented by means of different 
sets of elements, then the symbols {xdy} which correspond to these different forms 




139J AN INTRODUCTORY MEMOIR UPON QUANTICS. 225 

of the same quantic. are mere transformatioDs of each other, i.e. they become in virtue 
of the relations between the different sets of elements identical. 

9. What precedes is a return to and generalization of the method employed in the 
iirst part of the memoir published in the Camh. Math, Jour,, t, iv. [1845], and Camb. 
a,nd DiM. Math. Jour., t, I. [1846], under the title "On Linear Transformations," [13 
WLnd 14], and Crelle, t. xxx. [1846], under the title "M^moire sur les Hyperd^termi- 
iiants," [*16], and which I shall refer to as my original memoir. I there consider in 
fact the invariants of a quantic 

linear in regard to n sets each of them of m facients, and I represent the coefEcients 
of a term Wrj/tZt... by rst...; there is no diflSculty in seeing that a, 13 being any two 
<lifferent numbers out of the series 1, 2, ...m, the operation {^/sda;J is identical with the 
operation 

^S... last,., -jr^, 

^%vhere the summations refer to «, ty... which pass respectively from 1 to m, both inclu- 
^ve; and the condition that a function, assumed to be an invariant, i.e. to contain 
*jnly the coeflScients, may be reduced to zero by the operation {«/i9ajJ — a?^aj«» is of 
«30urse simply the condition that such function may be reduced to zero by the opera- 
tion {^/i^xj ; ^he condition in question is therefore the same thing as the equation 

cDf my original memoir. 

10. But the definition in the present memoir includes also the method made use 
CDf in the second part of my original memoir. This method is substantially as follows: 
^isonsider for simplicity a quantic £/' = 

^i^ntaining only the single set {x, y...), and let CT,, ^, ... be what the quantic becomes 
"Vrhen the set {x^y ...) is successively replaced by the sets (a:,, yi, ...), (ar„ yj,...), ... the 
Xiumber of these new sets being equal to or greater than the number of facients in 
t;he set. Suppose that A, B, C7, ... are any of the determinants 

^«ii ^af|> ^«,> ••• 
9yi> 9y»> 9y,> 



^hen forming the derivative 

'^here p, q, r ... are any positive integers, the function so obtained is a covariant in- 
xrolving the sets (a?i, yi,...)> (^f> yf>«««) ^m ^^^ if *ft®r *^h® diflferentiations we replace 

c. n. 29 



226 AN INTRODUCTORY MEMOIR UPON QU ANTICS. [l39 

these sets by the original set {x, y, ...)* ^^ hAwe a co variant involving only the original 
set (x, y, ...) and of course the coefficients of the quantic. It is in £act easy to show 
that any such derivative is a covariant according to the definition given in this 
Memoir. But to do this some preliminary explanations are necessary. 

11. I consider any two operations P, Q, involving each or either of them differeD- 
tiations in respect of variables contained in the other of them. It is required to 
investigate the effect of the operation P . Q, where the operation Q is to be in the 
first place performed upon some operand tl, and the operation P is then to be per- 
formed on the operand Qfl. Suppose that P involves the differentiations da, ^,... in 
respect of variables a, b, ... contained in Q and il, we must as usual in the operatioD 
P replace da, dj,,..- by da + d'a, 96 + d^6> ••• where the unaccentuated symbols operate 
only upon il, and the accentuated symbols operate only upon Q. Suppose that P is 
expanded in ascending powers of the symbols 3^., 9'6, ...f viz. in the form P-f-Pj + Pi + Ac., 
we have first to find the values of PjQ, PtQ, &c., by actually performing upon Q as 
operand the differentiations 9'a, 9'6--«- The symbols PQ, P,Q, P,Q, &c. will then contain 
only the differentiations d^, d^, ... which operate upon il, and the meaning of the ex- 
pression being once understood, we may write 

P.Q = PQ + P,Q + P,Q + &c. 

In particular if P be a linear function of 9a, dt, ..., we have to replace P by P + Pi, 
where P, is the same function of d'a, S't. ••• that P is of 3., 96, ..., and it is therefore 
clear that we have in this case 

P.Q = PQ + P(Q). 

where on the right-hand side in the term PQ the differentiations 9., 96,... are con- 
sidered as not in an}rwise affecting the symbol Q, while in the term P(Q) these 
differentiations, or what is the same thing, the operation P, is considered to be per- 
formed upon Q as operand. 

Again, if Q be a linear function of a, 6, c, ..., then PiQ = 0, PiQ = 0, Ac, and 
therefore P.Q = PQ-^PiQ\ and I shall in this case also (and consequently whenever 
P,Q = 0, P,Q = 0, &c.) write 

P.Q^PQ-^-PiQl 

where on the right-hand side in the term PQ the differentiations 9a, 96,... are con- 
sidered as not in anywise affecting the symbol Q, while the term P(Q) is in each case 
what has been in the first instance represented by PiQ. 

We have in like manner, if Q be a linear function of 9a, 96, 9c, ..., or if P be 
a linear function of a, 6, c, . . . , 

Q.P = QP + QiP); 
and from the two equations (since obviously PQ^QP) we derive 

P.Q-Q.P = P(Q)-Q(P), 
which is the form in which the equations are most frequently useful. 



139] AN INTRODUCTORY MEMOIR UPON QU ANTICS. 227 

12. I return to the expression 

and I suppose that after the dififerentiations the sets (or,, yi, ...), (^9, ys> •••)> ^^' ^^^ 
replaced by the original set (x, y, ...). To show that the result is a co variant, we must 
prove that it is reduced to zero by an operation 39 = 

It is easy to see that the change of the sets (x^, yi, ...), (x^ yji •••)» ^^ i^^ ^^^ original 

set (Xf y, ...) may be deferred until after the operation iD, provided that aSy is replaced 

by ^9y, + ^y,4-..., or if we please by Sxdy; we must therefore write ^ = {xidy} — Sxdy. 

Now in the equation 

il.B-B.il=il(B)-B(il), 

where, as before, A (W) denotes the result of the operation A performed upon iB as 
operand, and similarly ^(A) the result of the operation 39 performed upon A as 
operand, we see first that A (W) is a determinant two of the lines of which are 
identical, it is therefore equal to zero; and next, since 39 does not involve any 
differentiations affecting A, that 39 (A) is also equal to zero. Hence il . 39 ~ 39 . il = 
or A and 39 are convertible. But in like manner 39 is convertible with B, 0, &c., 
and consequently B is convertible with A^B^C^.... Now 391/itr,... =0; hence 

m.APB^C'...L\U,...^0, 

or ApB^O"... I/it/,... is a covariant, the proposition which was to be proved. 

13. I pass to a theorem which leads to another method of finding the covariants 
of a quantic. For this purpose I consider the quantic 



m m 



(♦$«?, y...\af, y'.. .)...), 

the coeflScients of which are mere numerical multiples of the elements (a, 6, c, ...); and 
in connexion with this quantic I consider the linear functions ^x-\-r)y..., fa/ + i;y"«> 
which treating (f, 17,...), (f, V»'«-)> *c- ®* coeflScients, may be represented in the form 

(f, 17, ...$ar, y, ...\ (f , 17', ...$a?', y', ...),... 
we may from the quantic (which for convenience I call U) form an operative quantic 

(♦$f, 17,... $f, V,...)---) 

(I call this quantic 6)» the coeflScients of which are mere numerical multiples of 
da, dt, 9«, •••> and which is such that 



ie. a product of powers of the linear functions. And it is to be remarked that as 
regards the quantic 6 and its covariants or other derivatives, the symbols da, df,, da... 
are to be considered as elements with respect to which we may differentiate, &c. 

29—2 



228 AN INTRODUCTORY MEMOIR UPON QUANTICS. [l39 

The quantic B gives rise to the symbols {^^], &c. analogous to the symbols {^y|, &c. 
formed from the quantic U. Suppose now that 4> is any quantic containing as well 
the coefficients as all or any of the sets of 6. Then [xdy] being a linear function of 
a, b, Ci... the variables to which the differentiations in ^ relate, we have 

again, [rfi^] being a linear function of the differentiations with respect to the variables^ 
da, dtt dc,... in 4>, we have 

these equations serve to show the meaning of the notations ^({^y}) and {n9(} (4>X an 
there exists between these symbols the singular equation 



14. The general demonstration of this equation presents no real difficulty, but toKn 
avoid the necessity of fixing upon a notation to distinguish the coefficients of the 
different terms and for the sake of simplicity, I shall merely exhibit by an exampl 
the principle of such general demonstration. Consider the quantic 

Cr= flur» + 36a;"^ + 3cy» + dy», 

this gives = pda + ^hfii, + ^rfde + rfd^ ; 

or if, for greater clearness, da, dt, 9c, da are represented by a, fiy 7, S, then 

and we have {aSy] = 369a + 2c96 4- dde, 

and {nd(} = 3a9^ + 2^89^ 4- 79a. 

Now considering 4> as a function of 9a, 9^, 9e, da, or, what is the same thing, ol 
a, A 7, S, we may write 

4> {{xdy}) = a> (36a + 2c/8 + c^) ; 

and if in the expression of <P we write a + 9a, ff-^-d^, 7 + 9c, 8 + 9^ for a, /8, 7, S (whe 
only the symbols 9a, d^, de, da are to be considered as affecting a, 6, c, ct as contain 
in the operand 36a + 2cl3 + dy), and reject the first term (or term independent o 
9a. 9^, 9c, 9rf in the expansion) we have the required value of 4>({d?Sy)}. This value is 

(9a4>9a + 9^4>96 + 9ya>9c)(36a + 2c/8 + *y); 

performing the differentiations 9a, 9^, de, da, the value is 

(3a9^ + 2/89y 4- 79«) *, 

i.e. we have * ({^y}) = {^«} (*)• 





139] AN INTRODUCTORY MEMOIR UPON QUANTICS. 229 

15. Suppose now that 4> is a covariant of 8, then the operation <I> performed 
upon any covariant of TJ gives rise to a covariant of the system 

(f, 17, ...$a:, y, ...), (f, V» •••$^» y'» •••)» &c. 

To prove this it is to be in the first instance noticed, that as regards (f, 17, ...$a?, y, ...), &c. 
we have [aody] = 178^, &c. Hence considering [xdy], &c. as referring to the quantic U, 
the operation 2 {aj0y} — aidy will be equivalent to [aidy] + 178^ — aSy, and therefore every 
covariant of the s)rstem must be reduced to zero by each of the operations 

iB = [xdy] + 179^ - a^y. 

This being the case, we have 

ID . ^ = B* 4- ID (^), 

^nations which it is obvious may be replaced by 

^ind consequently (in virtue of the theorem) by 

B.^ = B4> + ^3f(^), 

a>.iD = ^B + {i73f}(4>); 
^nd we have therefore 

B . a> - 4>.iD = - ({179^} - 173^)(4>) ; 

or, since 4> is a covariant of 0, we have iD . 4> = 4> . iD. And since every covariant 
of the system is reduced to zero by the operation 39, and therefore by the operation 
<>.19, such covariant will also be reduced to zero by the operation iD.<I>, or what is 
the same thing, the covariant operated on by 4>, is reduced to zero by the operation 
"9 and is therefore a covariant, i.e. <I> operating upon a covariant gives a covariant. 

16. In the case of a quantic such as {7 = 

(♦i^> y$^'» yO--X 

^e may instead of the new sets (f, ^), (f, rf)... employ the sets (y, — a?), (y', -a?')» &c- 
The operative quantic 8 is in this case defined by the equation 0Cr=O, and if 4> 
l)e, as before, any covariant of %, then 4> operating upon a covariant of U will give 
« covariant of U. The proof is nearly the same as in the preceding case ; we have 
instead of the equation ^({^y})= {179^} (*) the analogous equation 

Avhere on the left-hand side [aody] refers to TJy but on the right-hand side [acdy] refers 
to 0, and instead of ID = {aSy} 4- ^^ — aSy we have simply ID = {aSy} — aSy . 



230 AN INTRODUCTORY MEMOIR UPON QUANTICS. [139 

17. I pass next to the quantic 

which I shall in general consider under the form 

(a, b,...b\ a^'^x, y)"», 

but sometimes under the form 

(a, 6, ...6\ a'^x, y)** 

the former notation denoting, it will be remembered, 

and the latter notation 

But in particular cases the coefficients will be represented all of them by unacceu- 

tuated letters, thus (a, 6, c, dP^x, y)* will be used to denote CM;* + 36a;^-|-3ca:y* + (iy', 

and (a, 6, c, d^x, yY will be used to denote cwc* + fta;^ + cjcy* -f dy*, and so in all 
similar cases. 

Applying the general methods to the quantic 

(a, 6, ...6\ a'$a?, y)"*, 
we see that {y^*} = a96+ 269e...+m6'3a*, 

in fact, with these meanings of the symbols the quantic is reduced to zero by each 
of the operations {yS*} — y9«, {a;9y} — xdy ; hence according to the definition any function 
which is reduced to zero by each of the last-mentioned operations is a covariant of 
the quantic. But in accordance with a preceding remark, the covariant may be con- 
sidered as a rational and integral function, separately homogeneous in regard to the 
&cients {x, y) and the coefficients (a, 6, ...&\ a^). If instead of the single set (a;, y) 
the covariant contains the sets (^, y^), (^, y,), &c., then it must be reduced to zero 
by each of the operations ly9»} — /Sy9aj, {aSy}— /Sady (where Sy3,B = yi3,B, -fyiS^-l- ...), but 
I shall principally attend to the case in which the covariant contains only the set 

Suppose, for shortness, that the quantic is represented by U^ and let CTi, IT,,... 
be what U becomes when the set (x, y) is successively replaced by the sets {xi^ y^), 

(^i y%)i ^ Suppose moreover that 12 = 8^^8,^ — d^^dy,, &c., then the function 



12P13«23^... CTjlTjCr,..., 

in which, after the differentiations, the new sets {x^ yi), (a^, y^^... may be replaced 
by the original set (x^ y), will be a covariant of the quantic {7. And if the number 



39] AN INTRODUCTORY MEMOIR UPON QUANTIOS, 231 

f differentiations be such as to make the facients disappear, ie. if the sum of all 

he indices p, 9,... of the terms 12, &c. which contain the symbolic number 1, the 
am of all the indices p, r, ... of the terms which contain the symbolic number 2, 
nd so on, be severally equal to the degree of the quantic, we have an invariant, 
lie operative quantic becomes in the case under consideration 

he signs being alternately positive and negative; in fact it is easy to verify that this 
xpression gives identically OCT^O, and any co variant of operating on a covariant 
•f U gives rise to a covariant of U, 

18. But the quantic 

(a, 6, . . . b\ a'^x, y)*", 

onsidered as decomposable into linear factors, ie. as expressible in the form 

a(a?-ay)(a?-^y)..., 
[ives rise to a fresh series of results. We have in this case 

{y3*}= 3.4- dp..., 

{xdy} = - (a + fi...) ada + a«. + /ff©^ + ... ; 

n fact with these meanings of the symbc4s the quantic is reduced to zero by each 
)f the operations {ady} — aSy, {y9«} — y3aj, and we have consequently the definition of 
he covariant of a quantic considered as expressed in the form a(a? — ay)(a? — /3y),... 
\jid it will be remembered that these and the former values of the symbols [xdy] and 
ydg] are, when the same quantic is considered as represented under the two forms 
a, 6, . . .i\ a'$a?, yy* and a (a? — ay) (a? — I3y). . . , identical 

19. Consider now the expression 

a« (^ - ayy (a: - i8y)*...(a - i8)P... , 

¥here the sum of the indices j, p,.., of all the simple &ctors which contain a, the 
lum of the indices k, p,... of all the simple factors which contain fi, Sec are respec- 
ively equal to the index of the coefficient a. The index d and the indices p, &c. 
nay be considered as arbitrary, nevertheless within such limits as will give positive 
ralues (0 inclusive) for the indices j, k,.,,. 

The expression in question is reduced to zero by each of the operations 
xdy] — xdy, [yds] — yd^ ; and this is of course also the case with the expressions 
)btained by interchanging in any manner the roots a, fi, 7,..., and therefore with 
the expression 

a«2(a?-ay)i(a?-/3y)»...(a-^)p..., 

where 2 denotes a summation with respect to all the different permutations of the 
roots a, fi, ... . 



232 AN INTEODUCrrORY MEMOIR UPON QU ANTICS. [139 

The function so obtained (which is of course a rational function of (a, 6, ...V, a)) 
will be a covariant, and if we suppose /i = md — 25/>, where Sp denotes the sum of all 
the indices p of the different terms (a — fiy, &c., then the covariant will be of the 
order fi (Le. of the degree fi in the facients x^ y), and of the degree in the co- 
efficienta 

20. In connexion with this covariant 

a* 2 (a? ~ ay)P (a; - i8y)*... (a - /3)P..., 

of the order fi and of the degree in the coefHcients, of the quantic C/'= 

a(a?~ay)(a?-^y)..., 
consider the covariant 

of a quantic F= 

in. which, after the differentiations, the sets (xi, yi), (x^, yj,), ... are replaced by the 
original set (a?, y). The last-mentioned covariant will be of the order m (^ — ^) 4- /*, 
and will be of the degree m in the coeflScients; and in particular if ^ = 5, i.e. if V 
be a quantic of the order d, then the covariant will be of the order fA and of the 
degree m in the coefficients. Hence to a covariant of the degree in the coefficients, 
of a quantic of the order m, there corresponds a covariant of the degree m in the 
coefficients, of a quantic of the order 0] the two covariants in question being each 
of them of the same order /t. And it is proper to notice, that if we had commenced 
with the covariant of the quantic F, a reverse process would have led to the 
covariant of the quantic U. We may, therefore, say that the covariants of a given 
order and of the degree in the coefficients, of a quantic of the order m, correspond 
each to each with the covariants of the same order and of the degree m in the 
coefficients, of a quantic of the order 0; and in particular the invariants of the degree 
d of a quantic of the order m, correspond each to each with the invariants of the 
degree ni of a quantic of the order 0. This is the law of reciprocity demonstrated 
by M. Hermite, by a method which (I am inclined to think) is substantially identical 
with that here made use of, although presented in a very different form: the dis- 
covery of the law, considered as a law relating to the number of invariants, is due 
to Mr Sylvester. The precise meaning of the law, in the last-mentioned point of 
view, requires some explanation. Suppose that we know all the really independent 
invariants of a quantic of the order m, the law gives the number of invariants of 
the degree m of a quantic of the order (it is convenient to assume > m), viz. of 
the invariants of the degree in question, which are linearly independent, or asyzygetic, 
Le. such that there do not exist any merely numerical multiples of these invariants 
having the sum zero; but the invariants in question may and in general will be 
connected inter se and with the other invariants of the quantic to which they belong 
by non-linear equations : and in particular the sjrstem of invariants of the degree m 
will comprise all the invariants of that degree (if any) which are rational and integral 



139] AN INTRODUCTORY MEMOIR UPON QUANTICS. 233 

functions of the invariants of lower degrees. The like observations apply to the system 
of covariants of a given order and of the degree m in the coefficients, of a quantic 
of the order ft 

21. The number of the really independent covariants of a quantic (♦$«?, y)"* is 
precisely equal to the order m of the quantic, Le. any covariant is a function 
(generally an irrational function only expressible as the root of an equation) of any 
m independent covariants, and in like manner the number of really independent in- 
variants is m — 2 ; we may, if we please, take m — 2 really independent invariants as 
part of the system of the m independent covariants; the quantic itself may be taken 
as one of the other two covariants, and any other covariant as the other of the two 
covariants; we may therefore say that every covariant is a function (generally an 
irrational function only expressible as the root of an equation) of m — 2 invariants, of 
the quantic itself and of a given covariant. 

22. Consider any covariant of the quantic 

(a, 6, ... b\ a'Jx, yf, 

and let this be of the order /i, and of the degree in the coefficients. It is very 
easily shown that md — fi is necessarily even. In particular in the case of an invariant 
(Le. when /t = 0) m0 is necessarily even^: so that a quantic of an odd order admits 
only of invariants of an even degree. But there is an important distinction between 
the cases of md^fi evenly even and oddly even. In the former case the covariant 
remains unaltered by the substitution of (y, x), (a\ b\ ... b, a) for {x, y), (a, b, ... 6\ a'); 
in the latter case the effect of the substitution is to change the sign of the covariant. 
The covariant may in the former case be called a symmetric covariant, and in the 
latter case a skew covariant. It may be noticed in passing, that the simplest skew 
invariant is M. Hermit e*s invariant of the degree 18 of a quantic of the order 5. 

23. There is another very simple condition which is satisfied by every covariant 
of the quantic 

(a, b,...b\ a^x, y)*", 

viz. if we consider the facients (x, y) as being respectively of the weights i, — i, and 
the coefficients (a, 6, ...6\ a) as being respectively of the weights — ^m, — ^+1, 
...^m — 1, ^m, then the weight of each term of the covariant will be zero. This is 
the most elegant statement of the law, but to avoid negative quantities, the state- 
ment may be modified as follows: — if the facients (x, y) are considered as being of 
the weights 1, respectively, and the coefficients (a, b...V, a') as being of the weights 
0, 1, ...,m — 1, m respectively, then the weight of each term of the covariant will be 

i(wi^ + /*)- 

^ I may remark that it was only M. Hermite^s important discovery of an invariant of the degree 18 of 
a qnantio of the order 5, which removed an erroneous impression which I had been under from the oom- 
mencement of the subject, that mO was of necessity evenly even. 

C. II. 80 



234 AN INTRODUCTORY MEMOIR UPON QU ANTICS. [l39 

24. The preceding laws as to the form of a covariant have been stated here by 
way of anticipation, principally for the sake of the remark, that they so tso' define the 
form of a covariant as to render it in very many cases practicable with a moderate 
amount of labour to complete the investigations by means of the operation {ady} — xd^ 
and {ydg] — yd^^ In fact, for finding the covariants of a given order, and of a given 
degree in the coefficients, we may form the most general function of the proper order 
and degree in the coefficients, satisfying the prescribed conditions as to symmetry and 
weight: such function, if reduced to zero by one of the operations in question, will, 
on accoimt of the symmetry, be reduced to zero by the other of the operations in 
question; it is therefore only necessary to effect upon it, e.g. the operation {ody} — a:9y, 
and to determine if possible the indeterminate coefficients in such manner as to 
render the result identically zero: of course when this cannot be done there is not 
any covariant of the form in question. It is moreover proper to remark, as regards 
invariants, that if an invariant be expanded in a series of ascending powers of the 
first coefficient a, and the first term of the expansion is known, all the remaining 
terms can be at once deduced by mere differentiations. There is one very important 
case in which the value of such first term (i.e. the value of the invariant when a is 
put equal to 0) can be deduced firom the corresponding invariant of a quantic of the 
next inferior order; the case in question is that of the discriminant (or function 
which equated to zero expresses the equality of a pair of roots); for by Joachimsthars 
theorem, if in the discriminant of the quantic (a, b, ... 6\ a^$a?, y)^ we write a = 0, the 
result contains 6* as a factor, and divested of this factor is precisely the discriminant 
of the quantic of the order m — 1 obtained firom the given quantic by writing a = 
and throwing out the &ctor x: this is in practice a very convenient method for the 
calculation of the discriminants of quantics of successive orders. It is also to be 
noticed as regards covariants, that when the first or last coefficient of any covariant 
(i.e. the coefficient of the highest power of either of the facients) is known, all the 
other coefficients can be deduced by mere differentiations. 

Postscript added October 7th, 1854. — I have, since the preceding memoir was 
written, found with respect to the covariants of a quantic (♦ $a7, y)"*, that a function 
of any order and degree in the coefficients satisfying the necessary condition as to 
weight, and such that it is reduced to zero by one of the operations {xdy} — xdy, 
[ydxl—ydzf will of necessity be reduced to zero by the other of the two operations, 
i.e. it will be a covariant; and I have been thereby led to the discovery of the law 
for the number of asyzygetic covariants of a given order and degree in the coefficients; 
from this law I deduce as a corollary, the law of reciprocity of MM. Sylvester and 
Hermite. I hope to return to the subject in a subsequent memoir. 



140] 



235 



140. 



RESEARCHES ON THE PARTITION OF NUMBERS. 

[From the Philosophical Transactions of the Royal Society of London, vol. cxlv for the 
year 1855, pp. 127—140. Received April 14,— Read May 24, 1855.] 

I PROPOSE to discuss the following problem : " To find in how many ways a 
number q can be made up of the elements a, 6, c, ... each element being repeatable 
an indefinite number of times." The required number of partitions is represented by 
the notation 

P(a, 6, c, ...)?, 

and we have, as is well known, 

P{a,b.c, ...)3 = coefficient afl in (- i_^)(i _^)(i _^)... . 
where the expansion is to be effected in ascending powers of x. 

It may be as well to remark that each element is to be considered as a separate 
and distinct element, notwithstanding any equalities which may exist between the 
numbers a, 6, c, . . . ; thus, although a = 6, yet a -f a + a -h &c. and a 4- a -f 6 -f &c. are to 
be considered as two different partitions of the number q, and so in all similar cases. 

The solution of the problem is thus seen to depend upon the theory, to which I 
now proceed, of the expansion of algebraical firactions. 

Consider an algebraical fraction v , 

jx 

where the denominator is the product of any number of factors (the same or different) 
of the form 1 — a^**. Suppose in general that [1 — a^] denotes the irreducible factor of 
1 — af^, L e. the factor which, equated to zero, gives the prime roots of the equation 
1 - a:^ = 0. We have 

30—2 



236 RE8EARCHBS ON THE PARTITION OF NUMBEB& [l40 

where m' denotes any divisor whatever of m (unity and the number m itself not 
excluded). Hence, if a represent a divisor of one or more of the indices m, and h 
be the number of the indices of which a is a divisor, we have 

/a? «n[l -«•]*. 

Now considering apart from the others one of the multiple £EU^rs [! — «•]*, we 
may write /a: = [1 - «•]*/«• 



^ is decomposed 






4- Ac, 

where I{x) denotes the integral part, and the &c. refers to the fractional terms 
depending upon the other multiple factors such as [1 — ««]* The functions Qx are 
to be considered as functions with indeterminate coefficients, the degree of each such 
frmction being inferior by unity to that of the corresponding denominator; and it is 
proper to remark that the number of the indeterminate coefficients in all the frmctions 
6x together is equal to the degree of the denominator fx, 

dx 
The term (aS,)*"' rf-T^ ™*y ^ reduced to the form 

qx g,x 

[i -"^ "^ [1 - a^]*-> "^ ^^'^ 

the functions gx being of the same degree as Ox, and the coefficients of these functions 
being linearly connected with those of the function 0x, The first of the foregoing 
terms is the only term on the right-hand side which contains the denominator [1 — a;"]* ; 
hence, multiplying by this denominator and then writing [1 — a;*] = 0, we find 

ihx 

which is true when x is any root whatever of the equation [1 — a;"] = 0. Now by 

means of the equation [1 — a;^] = 0, j~ may be expressed in the form of a rational and 

integral function Ox, the degree of which is less by unity than that of [1— a^]. We 
have therefore Gx=^gx, an equation which is satisfied by each root of [1— a^] = 0, 
and which is therefore an identical equation ; gx is thus determined, and the coefficients 
of Ox being linear functions of those of gx, the function Ox may be considered as 
determined And this being so, the function 



fx ^-" [l-af^] 



140] BESEAKCHES ON THE PARTITION OF NUMBERS. 237 

will be a fraction the denominator of which does not contain any power of [1 — af^] 
higher than [1 — «*]*"* ; and therefore 0iX can be found in the same way as 0x, and 
similarly OtX, and so on. And the fractional parts being determined, the integral part 

may be found by subtracting from ^ the sum of the fractional parts, so that the fraction 
J- can by a direct process be decomposed in the above-mentioned form. 

Particular terms in the decomposition of certain fractions may be obtained with 
great facility. Thus m being a prime number, assume 

1 St ^ ^^ 

= &c. -f 



then observing that (1 — aJ^) = (l — ip)[l — a?^], we have for [1— a?^] = 0, 



5a; = 



Now u being any quantity whatever and x being a root of [1 — x^"] = 0, we have 
identically 

[1 - u'**] =(t4 - x)(u-a?) ... (m - 3^^) ; 

and therefore putting ti = l, we have m = (l— a?) (1 — «•)... (1— a;*^*), and therefore 

5a;=-, 
m 

whence 

1 _o 1 1 

Again, m being as before a prime number, assume 

= &c. 4- 



{l-x){\-a?)...{l-af^) ' [l-a;"']' 

we have in this case for [1 — af^] = 0, 

0x^ 



which is immediately reduced to 5a? = — , . Now 

'' ml—x 

tt — a? u — x ^ ^ 

or putting u » 1, 

= (TO-l)+(m-2)a;...+a;«-»; 



l-x 



238 RESEARCHES ON THE PARTITION OF NUMBERS. [l40 

aud substituting this in the value of Oxy we find 

1 « J_ (m - 1) + (tw - 2)a? ... + af^ 

(l-a:)(l-a;«)...(l-a?'~)'" ■*"m« [l-aJ~] 

The preceding decomposition of the fraction ?- gives very readily the expansion ot 
the fraction in ascending powers of x. For, consider a fiuction such as 

0x 

where the degree of the numerator is in general less by unity than that of the 
denominator ; we have 

l-a;« = [l-a:«]n[l-a:«'], 

. where a' denotes any divisor of a (including unity, but not including the number a 
itself). The fraction may therefore be written under the form 

0xTl [1 - Qf"'] 

where the degree of the numerator is in general less by unity than that of the 
denominator, Le. is equal to a — 1. Suppose that b is any divisor of a (including 
unity, but not including the number a itself), then 1 — a:* is a divisor of H [1 — sfi^, and 
therefore of the numerator of the fraction. Hence representing this numerator by 

A^ + A^x ... + Aa-iOf^', 

and putting a = 5c, we have (corresponding to the case 6 = 1) 

^0 "H "^i + A^ ... + A^^i ^ 0, 
and generally for the divisor 6, 

Ai + Aif^i . . . + -4 (e_n6+i = 0, 

* 

^6—1 + <4s6~i . . . + -^efr-i = 0. 

Suppose now that a^ denotes a circulating element to the period a, Le. write 

a, = 1, 9 = (mod. a), 

a, = in everj' other case; 

a frmction such as 

A^ + AxQ^\ . . . + Af^ia^^^^i 

will bo a circulating ftuiction, or circulator to the period a, and may be represented 
by the notation 

(A^y ill, •••'^•^ circlor o^. 



140] RESEARCHES ON THE PARTITION 01^ NUMBERS. 239 

In the case however where the coefficients A satisfy, for each divisor b of the number 
a, the above-mentioned equations, the circulating function is what I call a prime 
circulator, and I represent it by the notation 

(^0, ^i, ,,, ^(g^i) per Ciq. 

By means of this notation we have at once 

0x 
coefficient afl in tt- — -^1= (-^o, Ai...Aa^i) per aq, 

and thence also 

0x 
coefficient a?^ in (aidxY n — ;;ii ~ 5''^('^o» -^i-^-^a-i) P^r aq. 

Hence assuming that in the fraction ^ the degree of the numerator is less than that 
of the denominator (so that there is not any integral part), we have 

coefficient a^ in ^ = X q^(Ao, A^, ..,Aa^i) per a^; 

or, if we wish to put in evidence the non-circulating part arising from the divisor a = 1, 

coefficient .^^ in ^ = A^-^ + B^ ...^-Lq^-M 

+ 2 /(A, A^...Aa-^) per a,; 

where k denotes the number of the factors 1 — a?"* in the denominator fx, a is any 
divisor (unity excluded) of one or more of the indices m; and for each value of- a 
r extends from r = to r = & — 1, where k denotes the number of indices m of which 
a is a divisor. The particular results previously obtained show, that m being a 
prime number, 

coefficient a^ in (i ,^)(i ^^) ^,, (i ,^^ = &c.4-^( 1,-1, 0, 0, ...) F^ ^g> 

and 

coefficient ^ in (i _^)(i _^)...(t _^„) = &c. + ^,(m-l,-l,-l. ...) perm,. 

Suppose, as before, that the degree of ^ is less than that of fx, and let the 
analytical expression above obtained for the coefficient of a;? in the expansion in 

^)X 

ascending powers of x of the fraction ^ be represented by Fq, it is very remarkable 

(hx 
that if we expand -^ in descending powers of a?, then the coefficient of afl in this 

new expansion {q is here of course negative, since the expansion contains only 
negative powers of x) is precisely equal to —Fq; this is in fact at once seen to be 



240 RESEARCHES ON THE PARTITION OF NUMBERS. [140 

the case with respect to each of the partial fractions into which ^ has been de- 

composed, and it is consequently the case with respect to the fraction itself \ This 
gives rise to a result of some importance. Suppose that ^ and fx are respectively 

of the degrees J!V and 2); it is clear from the form of ^a? that we have /(-) = (— /a?"^; 
and I suppose that i>x is also such that ^ (-) =(±)'ic"'^^; then writing D^N^h, 

and supposing that ^ is expanded in descending powers of x, so that the coefficient 

of ^ in the expansion is ^Fq, it is in the first place clear that the expansion will 
commence with the term ar~*, and we must therefore have 

Fq^O 

for all values of q from } = — 1 to 9 = — (A — 1). 

Consider next the coefficient of a term ic~*~«, where 9 is or positive; the 
coefficient in question, the value of which is — ^(— A — 5), is obviously equal to the 

coefficient of a^+« in the expansion in ascending powers of x of — ~, Le. to 

(±)V/ coefficient a^+« in ^, 
. fx 

or what is the same thing, to 

(±y(-y coefficient 0^ in ^; 

jx 

and we have therefore, q being zero or positive, 

F(-h-q) = -(±n-yFq. 
In particular, when ^ = 1, Fq^O 

for all values of q from j = — 1 to 5 = — (D — 1) ; and q being or positive, 

F(^D^q)^{^r'Fq. 

The preceding investigations show the general form of the ftmction P(a, 6, c,...)?, 
viz. that 

P(a, b, c,...)3 = -4g*~^ + 5g'*-»...+Z5 + -If + 29^(^10, Ai,...Ai^i) per Ig, 

a formula in which k denotes the number of the elements a, b, c, ...^c., and I is 
any divisor (unity excluded) of one or more of these elements; the summation in the 
case of each such divisor extends from r = to r = A? — 1, where k is the number of 
the elements a, b, c, ...&c. of which Z is a divisor; and the investigations indicate 

^ The property is a fondamental one in the general theory of deyelopments. 



140] RESEARCHES ON THE PARTITION OF NUMBERS. 241 

how the values of the coefficients A of the prime circulators are to be obtained. It 
has been moreover in eflfect shown, that ifi) = a + 6 + c + ..., then, writing for shortness 
P(q) instead of P(a, b, c, ...)?> ^^ have 

P(q) = 
for all values of q from } = — 1 to 5^ = — (D — 1), and that q being or positive, 

P(-D-q) = i-r'P(q); 

these last theorems are however uninterpretable in the theory of partitions, and the}' 
apply only to the analytical expression for P(q). 

I have calculated the following particular results: — 

P(l. 2)y =i{2? + 3 

+ (1, -1) per 2,1 

P(l, 2, 3)9 =i^J6g» + 36? + 47 



72 



(. 



+ 9(1, -1) per 2g 
+ 8(2, -1, -1) per 3jl 

P(l, 2, 3. 4)9 =^|29» + 309' + 1353 + 175 

+ (9? + 45)(l, -1) per 2, 
+ 32 (1, 0, - 1) per 3, 
+ 36 (1, 0, -1, 0) per 4,1 

P(l, 2, 3, 4, 5) g = gg|^ JSO g« + 900 g* + 9300 5' + 38250 g + 50651 

+ (13503 + 10126) (1, -1) per 2, 
+ 3200 (2, - 1, - 1) per 3, 

+ 5400 (1, 1, - 1, - 1) per 4, 

+ 3466 (4, -1,-1, - 1, - 1) per 5, 

P(2)q =\[l 

+ (1, - 1) per 2,| 

^(2.3)9 =^{29 + 5 

+ 3 (1, - 1) per 2, 

+ 4(1, -1,0) per 3, 1 

C. II. 



31 



242 



RESEARCHES ON THE PARTITION OF NUMBERS. 



[140 



!"{% 3, 4)g 



288 



P(2, 3,4. 5)5 = 



P(2, 3, 4, 5, 6)3 = 



6 ^r' + 54 5 + 107 

+ (18g + 81)(l, -1) per 2, 
+ 32 (2, - 1, - 1) per 8, 
+ 36 (1, -1, -1,1) per 4,1 

+ (45} + 315)(l, -1) per 2, 
+ 160 (1, - 1. 0) per 3, 
+ 180 (1, 0, - 1, 0) per 4, 
+ 288 (1, - 1, 0, 0, 0) per 3,1 

[lO 3« + 4009» + 5550 3»+ 31000 q + 56877 

+ (450 3»+ 9000? + 39075) (1, -1) per 2, 

(1, - 1. 0) per 3, 

(21, - 19, - 2) per 3, 

(1, 0, - 1, 0) per 4, 

(4, - 1, - 1, - 1, - 1) per 5, 

(1, -1,-2,-1, 1. 2) per 6,1 



172800 



+ 3200 9 
+ 1600 
+ 10800 
+ 6912 
+ 4800 



Pil, 2, 3, o)q = 



JL_ ( 

720 



P(\, 2, 2, 3, 4)9 = 



4y' + 669"+ 3249 + 451 

+ 45 (1, - 1) per 2, 

+ 80 (1, - 1, 0) per 3, 

+ 144(1, 0, 0, 0, -1) per 5,1 

g^ je 9* + 144 9" + 1194 3» + 3960 } + 4267 

+ (64 9" + 648 9 + 1701) (1, -1) per 2, 
+ 256 (2, -1,-1) per 3, 

+ 432 (1, 0, -1, 0) per 4,1 



P(8)9 



^> 



+ 1 (1, -1) per 2, 

+ 2 (1, 0, - 1, 0) per 4, 

+ 8(1, 0, 0, 0, -1, 0, 0, 0) per 8,1 



140] RESEABCHES ON THE PARTITION OF NUMBERS. 243 

+ 7 (1, - 1) per 2, 

+ 14 (1. -1,-1, 1) per 4, 



which are. I think, worth preserving. 



+ 16 (3, 2, 1, 0, -1,-2,-3) per 7^ 
+ 66 (0, -1,-1, 0, 0, 1, 1, 0) per sj, 



Received April 14,— Read May 3 and 10, 1855. 

I proceed to discuss the following problem: "To find in how many ways a 
number q can be made up as a sum of m terms with the elements 0, 1, 2, . . . A;, 
each element being repeatable an indefinite number of timea" The required number 
of partitions is represented by 

P(0, 1, %...k)^q, 

and the number of partitions of q less the number of partitions of j — 1 is repre- 
sented by 

F(0, 1, 2, ...&)«}. 

We have, as is well known, 

P(0. 1. 2....A)»3 = coeffieient ^^ in (i _,)(i _i)...(i _^,) . 

where the expansion is to be effected in ascending powers of z. Now 
1 -i.lz^' (l-a^-^0(l-a^+«) , 

the general term being 

(1 - a^+0(l - a:*+») ... (1 - a^^"*) 
(l-a:)(l-a;»)... {l-af^) 

or, what is the same thing, 

(l-a:)(l-a;«)... (1-a?*) 
and consequently 

P(0. 1. 2. ...*)-.5 = eoeflBeient ^ in ^' (i-l^d -C)..r(i -S) ' 

to transform this expression I make use of the equation 

(l+a:^)(l+.;»^)...(l+.*^) = l+-L__^+ (\,^/(\^^) V + &C., 

31—2 



z^, 



s^. 



244 RESEARCHES ON THE PARTITION OF NUMBERS. [140 

where the general term is 






and the series is a finite one, the last term being that corresponding to s^k, viz. 
^(i+i)^ Writing —a^ for z, and substituting the resulting value of 

(I - a-H^i) (1 - -r~+«) . . . (1 - a-»+*) 
in the formula for P(0, 1, 2, ...4)"y, we have 

P(0. 1. 2. ... A)-v = S,^(-y coeffideot ^ in (i.^)(i_^)...(i_^)(l-^)(l-<'^)-.(l-x*n ' 

where the summation extends firom ^^0 to s^k; but if for any value of s between 
these limits «m + ^(«+l) becomes greater than q, then it is clear that the summation 
need only be extended firom 8=^0 to the last preceding value of s, or what is the 
same thing, firom « = to the greatest value of s for which j — «m — ^«(«+l) is 
positive or zenx 

It is obvious, that if y > km, then 

P(0, 1, 2...A)-}«0; 

and moreover, that if ^ > Ikm, then 

P(0, 1, 2,...it)~^«P(0, 1, 2,...it)~.itm-tf, 

so that we may always suppose j > Jibm. I write therefore q = ^(kni'-a) where a is 
xero or a {positive integer not greater than km, and is even or odd according as Arm 
18 oven or inld. Substituting this value of q and making a slight change in the 
form of the n^sult, we have 

whon« tht^ Hummation extends firom 9 = to the greatest value of s for which 
(U* -«)'H - ^ot- i«(^+ 1) IB positive or zero. But we may, if we please, consider the 
miiuiimtion m extending, when k is even, fix)m 8^0 to 8^\k — \, and when k is odd, 
\\\\\\\ /i-O to j» "• i (il-' 1); the terms corresponding to values of 8 greater than the 
gh»Hti»Hi value for which (iifc-«) m- Ja- J«(«4- 1) is positive or zero being of course 
equal to X(«rt»« It may be noticed, that the firaction will be a proper one if 
ri* (k H)(k-H'k-\)\ or substituting for 8 its greatest value, the firaction will be a 
iiiniirr Olio for all values of «, if, when it is even, o< Jifc(A: + 2), and when k is odd, 
«• \(k\ \)(k \ HV 

Wo have in a nimilar maimer, 

/••((). I. 2. ..Arv = coefficient «•,« in (-r -^)(l-^X(l -^*^) ' 



140] RESEABCHE8 ON THE PARTITION OF N(JMB£R8. 245 

which leads to 

^(0, 1, 2...ik)~i(Jtm-a) = 



where the summation extends, as in the former case, firom « = to the greatest value 
of «, for which (Jifc — «)m — ^a — i«(« + 1) is positive or zero, or, if we please, when k is 
even, from « = to « = iA?— 1, and when 8 is odd, from « = to « = J(A — 1). The 
condition, in order that the fruction may be a proper one for all values of s, is, 
when k is even, a + l<\k(k + 2), and when k is odd, a + 1 < i(A + l)(A: + 3). 

To transform the preceding expressions, I write when k is odd jc^ instead of x, 
and I put for shortness instead of ^k — s or 2(iA: — «), and y instead of Ja-h J«(«-»-l) 
or a + « (« + 1) ; we have to consider an expression of the form 

coefficient a^ in =r , 

where Fx is the product of factors of the form 1 — «:*. Suppose that a is the least 
common multiple of a and 0, then (1 — a*') -h (1 — «*) is an integral function of x, 
equal x^ suppose, and 1 -?- (1 — a;*) = ^^ -s- (1 - a?*'). Making this change in all the 
factors of Fx which require it (Le. in all the factors except those in which a is a 
multiple of 0), the general term becomes 

coefficient ar^ in —tz — , 

where Go; is a product of factors of the form 1 — af^'y in which a' is a multiple of 0, 
i.e. Go; is a rational and integral fiinction of of. But in the numerator a!*Hx we may 
reject, as not contributing to the formation of the coefficient of a^, all the terms in 
which the indices are not multiples of 0\ the numerator is thus reduced to a rational 
and integral function of afy and the general term is therefore of the form 

coefficient af^ in -Wv* 

or what is the same thing, of the form 

coefficient af* m — , 

icx 

where xx is the product of factors of the form 1 — ic*, and Xo; is a rational and integral 
function of x. The particular value of the fraction depends on the value of s: and 
uniting the dififerent terms, we have an expression 

\x 
coefficient x^ in 8m (—V — , 

^ KX 



^hich is equivalent to 



coefficient a"* in ^ , 

fx 



246 RESEARCHES ON THE PARTITION OF NUMBERS. [l40 

where /i; is a product of factors of the form 1 ---af^, and ^ is a rational and integral 
function of x. And it is clear that the fraction will be a proper one when each 
of the fractions in the original expression is a proper Auction, i.e. in the case of 
P(0, 1, 2...Jk)'»J(*^-«)» when for k even, a<iik(A:+2), and for k odd, a<i(Jfc+l)(it+3); 
and in the case of P'{0, 1, 2 ... A:)^ J(A?m — a), when for k even, a+l<\k(k + 2)y and 
for k odd, a+1 <l{k-\-l){k-hS). 

We see, therefore, that 

P(0, 1, 2...ifc)«»K*wi-aX 
and 

1^(0, 1, 2...ifc)«i(^-«X 
are each of them of the form 

coefficient af* in ?-, 
where fx is the product of factors of the form 1 — a*, and up to certain limiting values 

(dX 

of a the fiuction is a proper fraction. When the fraction ~- is known, we may there- 

fore obtain by the method employed in the former part of this Memoir, anal3rtical 
expressions (involving prime circulators) for the functions P and P', 

As an example, take 

P(0, 1, 2, 3)'^|m, 
which is equal to 

1 



coefficient a^ in 



— coefficient af* in 



(l-a;«)(l-a?*)(l-««) 



(l-aj«)(l-ic»)(l -«?•)• 
The multiplier for the first fraction is 

which is equal to 

1+ a;* -h ar* + ic« + 2a;» + a^' + «*'. 

Hence, rejecting in the numerator the terms the indices of which are not divisible 
by 3, the first term becomes 

coefficient ac^ in 



(l-a^)(l-aj»«)(l-a^)' 
or what is the same thing, the first term is 

1+a^ + a?* 



coefficient xl^ in 



(l-a^)»(l-a?*)' 



140] RESEARCHES ON THE PARTITION OF NUidBERS. 247 

and, the second term being 



— coefficient aS^ in 



(l-a:»)«(l-a?*)' 



1 +ip* 
we have P(0, 1, 2, 3)'"|w = coeflBcient a^ in /i ^a^vn --ar<V 

and similarly it may be shown, that 



P(0, 1, 2, 3)"* i(3w - 1) = coefficient af» in 



(l-ic«)»(l-ar*) 

As another example, take 

P'(0, 1, 2, 3, 4, 5)fm, 
which is equal to 

1 



coefficient a*^ in 



— coefficient a^ in 



+ coefficient off^ in 



(1 - a?*) (1 - a^) (1 - ic*) (1 - a?io) 

(^ 

(1 - a;«) (1 - «?•) (1 - a^) (1 - a:«) 



(1 - a;«) (1 - a?*) (1 - a;*) (1 - ««) • 
The multiplier for the first firaction is 

which is a function of a? of the order 36, the coefficients of which are 

1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 4, 4, 6, 4, 6, 5, 7, 5, 7, 5, 7, 5, 6, 4, 6, 4, 4, 3, 4. 2, 3, 1, 2, 1, 1, 0, 1, 
and the first part becomes therefore 

coefficient aS^ m — -p. ^r-r- — --r—- — —^—- — — — . 

(1 -a^)(l -a^)(l -aj'Xl -aJ®) 

The multiplier for the second fraction is 

(l-a^)(l-a^')(l-a;»*) 
(l-a;»)(l-aj*)(l -a^)' 

which is a function of a? of the order 14, the coefficients of which are 

1, 1, 2, 1, 3, 2, 3, 1, 3, 2, 3, 1, 2, 1, 1 ; 
and the second term becomes 

iX3- -L «• 2a^ + 2a:* + 3a^ + «" + a:" 
-coefficient a^ m (i ^^y^x -ar^)(l -a:") > 



248 RESEABCHES ON THE PARTITION OF NUMBERS. [UO 

and the third term is 

coefficient «** in 



Now the fractions may be reduced to a common denominator 

(1 -a;«)(l -a?*)(l -«•)(! -«!») 

1 — a^ 

by multipljring the terms of the second fraction by :. — — (= 1 + a;* + a?*), and the terras 

1 — «" 
of the third Auction by ^ ^ (= 1 + a?*) ; performing the operations and adding, the 

numerator and denominator of the resulting fraction will each of them contain the 
factor 1 — a^ ; and casting this out, we find 



P(0, 1, 2, 3, 4, o)** fm = coefficient ai^ in 



(l-a?*)(l-a;«)(l-a:»)' 



I have calculated by this method several other particular cases, which are given 
in my "Second Memoir upon Quantics", [141], the present researches were in fact 
made for the sake of their application to that theory. 

Received April 20,— Read May 3 and 10, 1855. 

Since the preceding portions of the present Memoir were written, Mr Sylvester 
has communicated to me a remarkable theorem which has led me to the following 
additional investigations ^ 

Let ^ be a rational fraction, and let (a? — a?i)* be a &ctor of the denominator fr, 
/« 

then if 






denote the portion which is made up of the simple fractions having powers of x — Xi 
for their denominators, we have by a known theorem 

. ^l = coefficient - in %j — : — ( . 

Now by a theorem of Jacobi's and Cauchy's, 

coefficient - in ^f: = coefficient - in F(yyt)'^'t] 

Z V 

whence, writing Xi + z=^ ^«~S we have 



J^l = coefficient - in -i ^; ^ . 

(/a:],, t x^-x^f(xi€r*) 



1 Mr Sylvester's researohes are published in the Quarterly Mathematical Journal^ July 1855, [vol. i. pp. 
141—152], and he has there given the general formula as well for the eiroulating as the non-dronlating part 
of the expression for the number of partitions.— Added 28rd February, 1856.— A. G. 



140] RESEARCHES ON THE PARTITION OF NUMBERS. 249 

Now putting for a moment x=^x^(f, we have 

1 1 1^1 

+ ^tf — 77 T\ + • • • » 



x^-x^ a:i(l-6'+*) iri(l-6') ' "^ x^{\-^) 
and 3^ = ar9jr, whence 



Xi^xtf Xi — x 1 Xi — x 1.2^ Xi — x 
the general term of which is 

fS-l 



Hence representing the general term of 

Xi<f> (XifT*) 

by x^i<~*, 80 that 

X^i = coefficient ^ m ^' f(x.e-*) ' 
we find, writing down only the general term, 



i/4*r *'* "^ n(*- 1) ("^^^U-^"^ ••• ' 



where the value of x^ depends upon that of «, and where 8 extends from « = 1 to « = Ar. 

Suppose now that the denominator is made up of £sictors (the same or different) 
of the form 1 — a?"*. And let a be any divisor of one or more of the indices m, 
and let k be the number of the indices of which a is a divisor. The denominator 
contains the divisor [1 — af^f, and consequently if p be any root of the equation 
[!—«"] = 0, the denominator contains the fetctor (p — a?)*. Hence writing p for Xi and 
taking the sum with respect to all the roots of the equation [1 — ic*] = 0, we find 

vhere vp = coefficient - in ^* jv^^v > 

and as before 8 extends from ^ = 1 to 8==k. We have thus the actual value of the 
function $x made use of in the memoir. 



A preceding formula gives 



coefficient t in S^\ f 



^Hich is a very simple expression for the non-circulating part of the fraction ^ 
^*^is is, in fact, Mr Sylvester's theorem above referred to. 

c. n. 32 



250 



[141 



141. 



A SECOND MEMOIR UPON QUANTICS. 

[From the Philosophical Transactiona of the Royal Society of London, vol. CXLVI. for the 
year, 1856, pp. 101—126. Received April 14,— Read May 24, 1855.] 

The present memoir is intended as a continuation of my Introductory Memoir 
upon Quantics, t. CXLIV. (1854), p. 245, and must be read in connexion with it ; the 
paragraphs of the two Memoirs are numbered continuously. The special subject of 
the present memoir is the theorem referred to in the Postscript to the Introductory 
Memoir, and the various developments arising thereout in relation to the number and 
form of the covariants of a binary quantic. 

25. I have already spoken of asyzygetic covariants and invariants, and I shall have 
occasion to speak of irreducible covariants and invariants. Considering in general a 
function u determined like a covariant or invariant by means of a system of partial 
differential equations, it will be convenient to explain what is meant by an asyzygetic 
integral and by an irreducible integral. Attending for greater simplicity only to a 
single set (a, 6, c, . . .), which in the case of the covariants or invariants of a single 
function will be as before the coefficients or elements of the function, it is assumed 
that the system admits of integrals of the form u^ P, u ^ Q, &c., or as we may 
express it, of integrals P, Q, &c., where P, Q, &c. are rational and integral homogeneous 
functions of the set (a, 6, c, ...), and moreover that the system is such that P, Q, &c. 
being integrals, <f>(P, Q, •*•) is also an integral Then considering only the integrals 
which are rational and integral homogeneous functions of the set (a, b, c, ...), integrals 
P, Q, JB, ... not connected by any linear equation or syzygy (such as \P + fiQ + vR ... 0),Q) 
are said to be asyzygetic; but in speaking of the asyzygetic integrals of a particular 
degree, it is implied that the integrals are a system such that every other integral of 

1 It is hardly necessary to remark, that the multipliers X, fi, r, ... , and generally any eoefficients or 
quantities not expressly stated to contain the set (a, b, c, ...), are considered as independent of the set, or 
to use a conyenient word, are considered as "trivials." 



141] A SECOND MEMOIR UPON QU ANTICS. 251 

the same degree can be expressed as a linear function (such as XP + /tQ + ]/i2...) of 
these integrals; and any integral P not expressible as a rational and integral homo- 
geneous fnnction of integrals of inferior degrees is said to be an irreducible integral 

26. Suppose now that A^^ A^, A^, &c. denote the number of asyzygetic integrals 
of the degrees 1, 2, 3, &c. respectively, and let a^, a,, a,, &c. be determined by the 
equations 

A = iai(ffi + l) + or„ 

-^8 = i «! («! + l)(ai + 2) + aia,-hor3, 

-44 = ^ai(ai+l)(«i+2)(ai + 3)-|-iai(ori-hl)a>-l-aia, + ia,(a,+ l)-ha4, &c., 

or what is the same thing, suppose that 

. l + A^x + A^-\-&c. =(l-a:)'^*(l-a;«)"^(l-a:»)"^...; 

a little consideration will show that a^, represents the number of irreducible integrals 
of the degree r less the number of linear relations or syzygies between the composite 
or non-irreducible integrals of the same degree. In £sict the asyzygetic integrals of 
the degree 1 are necessarily irreducible, i.e. Ai^a^. Represent for a moment the 
irreducible integrals of the degree 1 by X, X\ &c., then the composite integrals 
Z*, XX\ &c., the number of which is iai(ai + l), must be included among the asyzygetic 
integrals of the degree 2; and if the composite integrals in question were asyzygetic, 
there would remain ilj — i «!(«! + 1) for the number of irreducible integrals of the 
degree 2 ; but if there exist syzygies between the composite integrals in question, the 
number to be subtracted from A^ will be ^^^(ai + l) less the number of these syzygies, 
and we shall have -^a — i ai(ai + l), ie. ffj equal to the number of the irreducible 
integrals of the degree 2 less the number of syzygies between the composite integrals 
of the same degree. Again, suppose that Oj is negative = — A, we may for simplicity 
suppose that there are no irreducible integrals of the degree 2, but that the com- 
posite integrals of this degree, X^, XX', &c., are connected by fi^ syzygies, such as 
\X* + fiXX' + &C. = 0, XiZ* + fiiXX' + &c. = 0. The asyzygetic integrals of the degree 4 
include X\ X*X', Sec, the number of which is ^ Oi (ce, + 1) (Oi + 2) (a^ + 3) ; but these 
composite integrals are not asyzygetic, they are connected by syzygies which are 
angmentatives of the fi^ syzygies of the second degree, viz. by syzygies such as 

(XX« + fiZZ'...)Z» = 0, (XZ»-h/iZZ'...)ZZ' = 0, &c. (XiZ«+/AiZZ'...)Z« = 0, 

(XiZ» + /AiZZ'...)ZZ' = 0, &c., 

the number of which is iai(ai-hl))8j. And these syzygies are themselves not asyzygetic, 
they are connected by secondary syzygies such as 

Xj(VZ« + m-X'Z'...)Z» + /Ii(XZ« + /iaZZ'...)ZZ' + &c. 

-X(XiZ>+/iiZZ'...)Z»-fi(XiZ« + AhZZ'...)ZZ'-&c. = 0, &c. &c., 

32—2 



252 A SECOND MEMOm UPON QUANTIGS. [141 

the number of which is i)9s09, — 1). The real number of syzygies between the com- 
posite integrals X\ X*X\ &c. (Le. of the syzygies arising out of the Pt syzygies 
between X*, XX\ &c.), is therefore i ai(ai + 1)A — ii8,()8j— 1), and the number of 
integrals of the degree 4. arising out of the integrals and syzygies of the degrees 
1 and 2 respectively, is therefore 

or writing —a, instead of )8,, the number in question is 

A «!(«! + l)(or, + 2)(ai -h 3) + i ai(a, + 1) or, + i a, (a, + 1). 

The integrals of the degrees 1 and 3 give rise to ttiO, integrals of the degree 4; and if 
all the composite integnJs obtained as above were asjrzygetic, we should have 

^4-Affi(ai + l)(ai + 2)(aj + 3)-iai(ar,-hl)a,--ia,(a,-hl)-aia,, 

i.e. OL^ as the number of irreducible integrals of the degree 4; but if there exist any 
further syzygies between the composite integrals, then a^ will be the number of the 
irreducible integrals of the degree 4 less the number of such further syzygies, and the 
like reasoning is in all cases applicable. 

27. It may be remarked, that for any given partial differential equation, or system 
of such equations, there will be always a finite number v such that given v independent 
integrals every other integral is a function (in general an irrational fimction only 
expressible as the root of an equation) of the v independent integrals; and if to these 
integrals we join a single other integral not a rational function of the v integrals, it is 
easy to see that every other integral will be a rational function of the y + 1 integrals : 
but every such other integral will not in general be a rational and integral fimction of 
the ]/ + 1 integrals ; and lincorrecf] there is not in general any finite number whatever 
of integrals, such that every other integral is a rational and integral function of these 
integrals, Le. the number of irreducible integrals is in general infinite ; and it would seem 
that this is in fact the case in the theory of covarianta 

28. In the case of the co variants, or the invariants of a binary quantic, A^ \a given 
(this will appear in the sequel) as the coefficient of o;^ in the development, in ascending 

powers of a?, of a rational fi:uction ^ , where yi is of the form 

(1 - a:)^(l - «•)^ . .(1 - a:*y», 
and the degree of ^ is less than that of ^. We have therefore 

and consequently 

<^=(1 -a?)*-Xl-aj«y*-"-...(l -af'Y'^O -a*+')"**^'- • 



141] A SECOND MEMOIR UPON QU ANTICS. 253 

Now every rational factor of a binomial l—af^iB the irreducible factor of 1 — af^', 
where m is equal to or a submultiple of m. Hence in order that the series a^, or,, 3s»*-- 
may terminate, ^ must be made up of factors each of which is the irreducible factor 
of a binomial 1 — af^, or if ^ be itself irreducible, then <f>x must be the irreducible 
factor of a binomial 1 — a^. Conversely, if ^ be not of the form in question, the 
series a^, a,, a^, &c. will go on ad infinitum, and it is easy to see that there is no point 
in the series such that the terms beyond that point are all of them negative, i.e. there 
will be irreducible covariants or invariants of indefinitely high degrees; and the number 
of covariants or invariants will be infinite. The number . of invariants is first infinite in 
the case of a quantic of the seventh order, or septimic ; the number of covariants is first 
infinite in the case of a quantic of the fifth order, or quintic. [As is now well known, 
these conclusions are incorrect, the number of irreducible covariants or invariants is 
in every case finite.] 

'2d. Resuming the theory of binary quantics, I consider the quantic 

(a, 6,...6\ a^$a?, y)*". 
Here writing 

[ady] = mbda + (m — 1 ) cS^. . .+ a'9y, = )', 

any function which is reduced to zero by each of the operations X — yd^, Y—xdy is a 
CO variant of the quantic. But a co variant will always be considered as a rational 
and integral function separately homogeneous in regard to the facients {x, y) and to 
the coeflBcients (a, 6,...6\ d). And the words order and degree will be taken to refer 
to the facients and to the coeflBcients respectively. 

I commence by proving the theorem enunciated. No. 23. It follows at once from 
the definition, that the covariant is reduced to zero by the operation 



X — ydx . Y — xdy — Y— xdy . X — yd^t 
which is equivalent to 

X . F— Y ,X+ydy — aSx' 
Now 

X.Y = XY+X(Y) 

Y.X=YX+Y(X\ 

where XY and YX are equivalent operations, and 

X(F)= lmada-^2(m-l)bdt...+ mlb'dt', 

Y(X)= wl6aft...+ 2(m- ])b'dt-^\„ia:da, 

whence 

Jf (F)- F(Jf) = 7yia9a + (m — 2)63ft...-(m-2)6'9e> -ma'3a, =* suppose, 

and the covariant is therefore reduced to zero by the operation 

k + ydy — a^x- 
Now as regards a term a*b^..M^d''\a^y^, we have 

fc = wa + (i?i-2)/8..., '-(m-2)/3"-md 
ydy-aSx^'j-i; 



254 A SECOND MEMOIR UPON QUANTIC8. [141 

and we see at once that for each term of the covariant we must have 

ma + (m - 2) /8...- (m - 2) ;8^ - tna' + j - 1 = 0, 

i.e. if (x, y) are considered as being of the weights ^, —^ respectively, and (a, b,...b\ a) 
as being of the weights — |fn, — ^m + 1, ...^m— 1, ^m respectively, then the weight 
of each term of the covariant is zero. 

But if {x, y) are considered as being of the weights 1, respectively, and (a, 6,...6\ a) 
as being of the weights 0» l,...m — 1, m respectively, then writing the equation under 
the form 

and supposing that the covariant is of the order ^ and of the degree 0, each term of 
the covariant will be of the weight \ (mO + /i). 

I shall in the sequel consider the weight as reckoned in the last-mentioned manner. 
It is convenient to remark, that as regards any function of the coefficients of the degree 
6 and of the weight 9, we have 

X.Y-Y.X^md-iq. 

30. Consider now a covariant 

{A, B,...B, A^\x, yY 

of the order n and of the degree 0\ the covariant is reduced to zero by each of the 
operations X—yd,, T — aldy, and we are thus led to the systems of equations 

XA^O, 
XB = ijlA, 
XC = (jjL-l)B. 

Xff= 2(7. 

XA'=ff ; 
and 

YA = B, 

YB = 2(7, 

Y(T= 0* - 1) F, 
Yff= ,iA\ 
YA'= 0. 

Conversely if these equations are satisfied the function will be a covariant. 

I assume that il is a function of the degree and of the weight ^ (md — /i), satisfying 
the condition 



141] A SECX)ND MEMOm UPON QUANTICS. 255 

and I represent by YA, FM, Y^A, &c. the results obtained by successive operations 
with Y upon the function A, The function Y'A will be of the degree and of the 
weight ^(mO -- fi) + 8, And it is clear that in the series of terms YA, FM, FM, &e., 
we must at last come to a term which is equal to zero. In fact, since m is the 
greatest weight of any coeflBcient, the weight of F* is at most equal to m0, and therefore 
if i(md — /i)+«>m5, or 8 > ^(mO -^^ fi), we must have Y'A=0. 

Now writing for greater simplicity XY instead o{ X,Y, and so in similar cases, we 
have, as regards Y'A, 

XY-YX = fi-28. 



Hence 



^od consequently 



i*-5iinilarly 



^ therefore 



d again, 



therefore 



^nerally 



{XY-'YX)A = fiA, 
XYA=^YXA'\-lj^A=ij^A, 

(ZF- FX) F^ = (/i - 2) Fil, 

X Y^A = FZ Fil + (/i - 2 ) Fil 

= /aF4 + (a*- 2) F.4 = 2(/i- l)Fil. 

(XF-FZ)F«il=(/A-4)FM, 

XY^A = YXY^A + (/i - 4) Y^A 

= 2(/t- 1) F«il + 0i- 4)FM = 3 (/i- 2) FM, 

ZF'^=«(/A-« + l)FM. 



^ce putting « = ^+l, /i + 2, &c., we have 

ZF'*+M = 0, 

XYi'^^A = - (/i + 2) 1 . F'*+'^, 

ZF'*+M=-(/i + 3)2.F'*-«J, 
&c, 

F^+M = ; 

y^*^ unless this be so, La if F''+*-4 + 0, then from the second equation Z F'^+M + 0, and 
^^^fore Yf'^A^O, from the third equation ZF'*+»=NO, and therefore Z'^+M^O, and so 
^ dd infinitum^ Le. we must have Y'^'^^A = 0. 



^^^^^ations which show that 



256 A SECOND MEMOIR UPON QUANTIC8. [141 

81. The Buppoeitiona which have been made as to the function A, give therefore 
the equations 

XA =0. 

XYA =,iA. 

XY*A = 2(ji-\)YA, 

XYi^A'=(tYi'*'A, 
Yi'+'A = ; 
and if we now assume 

the sratem becomes 

XA'=0, 

XB = fiA, 

XC = (ji-l)B. 

J A' = F, 
YA' = 0; 

80 that the entire system of equations which express that (A, B.,.B, ^^^^f vY ^ 
a covariant is satisfied ; hence 

Theorem. Given a quantic (a, 6, ...6\ a'$a?, y)***; if il be a function of the 
coefficients of the degree and of the weight ^ (mO — fi) satisfying the condition 
XA =0, and if -B, C, ... jB", -4' are determined by the equations 

B^YA, C = ^YB,,..A' = ^YB\ 

then will 

(A, B,...B, A\x, yY 

be a covariant. 

In particular, a function A of the degree and of the weight ^m^, satisfying the 
condition XA = 0, will (also satisfy the equation YA = and vdll) be an invariant. 

32. I take now for A the most general function of the coefficients, of the degree 6 
and of the weight \ {mO — /x) ; then XA is a function of the degree 6 and of the weight 
^(m^ — /i)— 1, and the arbitrary coefficients in the function A are to be determined 
so that XA =■ 0. The number of arbitrary coefficients is equal to the number of 
terms in A^ and the number of the equations to be satisfied is equal to the number of 
terms in XA ; hence the number of the arbitrary coefficients which remains indeter- 
minate is equal to the number of terms in A less the number of terms in XA ; and 
since the covariant is completely determined when the leading coefficient is known, 



141] 



A SECOND MEMOIR UPON QUANTICS. 



257 



the diflference in question is equal to the number of the asyzygetic covariants, i.e. the 
number of the asyzygetic covariants of the order fi and the degree is equal to the 
number of terms of the degree and weight \ (mO — fi), less the number of terms of 
the degree d and weight J(md— /a)— 1. 

83. I shall now give some instances of the calculation of covariants by the method 
just explained. It is very convenient for this purpose to commence by forming the 
literal parts by Arbogast's Method of Derivations : we thus form tables such as the 
following :— 





a 


b 


c 










a* 


ah 


ac 
6« 


be 


6» 








a 


b 


c 


' ! 



ah 



6« 



ad 
be 



bd 



ed 



d" ! 



rt* 


a^b 


a*c 


aH 


abd 


aed 


cuP 


bd" 


ed" 


(P 






ah^ 


ahe 

b" 


ac" 
b^e 


b^d 
be" 


bed 


c'd 







a* 


M 


a»<J 


aH 


a^bd 


a^ed 


a^d" 


abcP 


a^<P 


ad^ 


bd^ 


ccP 


d* 






a«6» 


a^be 


a^c" 


ab'd 


abed 


ac'd 


b^d" 


bcd^ 


c^rf* 












ab* 


ah^c 
6* 


abc" 
b^c 


bH 
6V 


b^ed 
be" 


bc'd 


eH 









a 


b 


c 


d 


e 



a» 


ab 


ac 


ad 


M 


be 


bd 


ed 


d' 






V 


he 


bd 


ed 


<? 







C. II. 



33 



258 



A SECOND MEMOIB UPON QUANTICS. 



[141 



a' 


a*6 


a^c 


a^d 


a«« 


abe 


ace 


ode 


oe* 


W 


ce» 


de* 


^ 






ah* 


abc 


abd 


acd 


cuP 


bee 


bde 


cde 


d^. 












&» 




b*d 
be" 


bh 
bed 


c'd 




d^ 




- 
















<? 















34. Thus in the case of a cubic (a, 6, c, d$a?, y)*, the tables show that there will 
be a single invariant of the degree 4. Represent this by 

•\' Babcd 
4- Cad" 
+ Dl^d 

which is to be operated upon with a36 + 263c + 3c9d- This gives 



Le. 
E^ 



+ B 








+ 6i4 




a^cd 


+ 3/> 




+ 25 








ab\l 


+ '2E 




+ 6C 

+ 45 

putting . 




+ 35 
+ 35 




abc* 
b'c 


5 = 0, &c.; 


or 


zl 


= 1, we 


fi 


nd 5 



= -(], C=4, Z> = 4. 



— 3, and the invariant is 

— 6a6cd 
+ 4ac» 
+ 46»d 
-36V. 

Again, there is a covariant of the order 3 and the degree 3. The coefficient of .r* 
leading coefficient is 

Aa*d 
•i- Babe 

which operated upon with a3e> + 2ft3c4- 3c9d, gives 



+ 5 

+ 3C 

1 


+ 2Z? 


+ 3i4 


a'e 
ab' 



141] 



A SECOND MEMOIR UPON QUANTICS. 



259 



i.e. B-^SA =0, 3C+aB = 0; or putting ^ = 1, we have 5 = -3, C = 2, and the leading 
coefficient is 

- Sabc 
+ 2 6». 

The coefficient of ah/ is found by operating upon this with (369a + 2cdi, + ddc), this 
gives 

abd 

ac' 

b'c 



+ 6 
-9 


-6 
+ 12 


-3 



i.e. the required coefficient of a^y is 



Sabd 
- 6ac» 
4-3 6»c; 



and by operating upon this with J (369a + ^cdb + dde\ we have for the coefficient of a?y^ 



acd 
b'd 

be' 



4. » 
+ If 

-9 


+ 3 
+ 6 


-6 

4. » 



i.e. the coefficient of xi/* is 



— 3 acd 
+ 66»d 
~3 6c». 



Finally, operating upon this with ^ (369a + 2c9ft + d9c)> we have for the coefficient of y*, 







-1 


-3 


+ 8 
-2 


-2 



i.e. the coefficient of y* is 



and the covariant is 



+ 36cd 
-2c», 



ad' 
bed 



( abe-3 
V +2 


oc* -6 
6*c +3 


h*d +6 
ftc" -3 


1 

6e(^ + 3 
c» -2 



$a^» yf 



[I now write the numerical coefficients after instead of before the literal terms.] 

33—2 



260 



A SECOND MEMOIR UPON QU ANTICS. 



[141 



I have worked out the example in detail as a specimen of the most convenient method 
for the actual calculation of more complicated covaiiants^ 

35. The number of terms of the degree and of the weight q is obviously 
equal to the number of wajrs in which q can be made up as a sum of 6 terms 
with the elements (0, 1, 2, ...m), a number which is equal to the coeflBcient of aflg^ in 
the development of 

___^ 1 

and the number of the asyzygetic covariants of any particular degree for the quantic 
(♦$^> y)** can therefore be determined by means of this development. In the case of 
a cubic, for example, the function to be developed is 



{\ " z){l - xz) {\ -a^z) {I -a?zy 
which is equal to 

where the coeflScients are given by the following table ; on account of the symmetry, 
the series of coefficients for each power of ^r is continued only to the middle tenn or 
middle of the series. 











1 






1 


1 








1 


1 


2 


2 






1 


1 


2 


3 


3 








1 




1 




2 


3 


4 


4 


5 




1 




I 


2 


3 


4 


5 


6 


6 




1 


1 


2 


3 


4 


5 


7 


7 


8 


8 



(0) 

(1) 

(2) 
(3) 
(4) 
(5) 
(6) 



^ Note added Feb. 7, 1856.— The following method for the calculation of an invariant or of the leading 
coefficient of a covariant, is easily seen to be identical in principle with that given in the text Write down 
all the terms of the weight next inferior to that of the invariant or leading coefficient, and operate on each 
of theee separately with the symbol 

ind. 6 . - + 2 ind. c . r +...(»»- 1) ind. 6* . -^ , 

where we are first to multiply by the fraction, rejecting negative powers, and then by the index of the proper 
letter in the term so obtained. Equating the results to zero, we obtain equations between the terms of the 
invariant or leading coefficient, and replacing in these equations each term by its numerical coefficient in the 



141] 



A SECOND MEMOIR UPON QUANTICS. 



261 



and from this, by subtracting from each coefficient the coefficient which immediately 
precedes it, we form the table: 



(0) 















L 








1 













1 





1 








1 





1 


1 













1 





1 


1 


1 





1 






1 





1 


1 


1 


1 


1 


( 


} 




1 





1 


1 


1 


1 


2 





1 












































(2) 
(3) 

(5) 



The successive lines fix the number and character of the covariants of the degrees 
0, 1, 2, 3, &C. The line (0), if this were to be interpreted, would show that there is a 
single covariant of the degree ; this covariant is of course merely the absolute con- 
stant unity, and may be excluded. The line (1) shows that there is a single covariant 
of the degree 1, viz. a covariant of the order 3; this is the cubic itself, which I 
represent by U. The line (2) shows that there are two asyzygetic covariants of the 
degree 2, viz. one of the order 6, this is merely JJ\ and one of the order 2, this I 
represent by H, The line (3) shows that there are three asyzygetic covariants of the 
degree 3, viz. one of the order 9, this is JJ* ; one of the order 5, this is UH, and one of 
the order 3, this I represent by 4). The line (4) shows that there are five asyzygetic 
covariants of the degree 4, viz. one of the order 12, this is U^\ one of the order 8, 
this is V^H\ one of the order 6, this is JT*; and one of the order 0, ie. an invariant, 
this I represent by V. The line (5) shows that there are six asyzygetic covariants of 
the degree 5, viz. one of the order 15, this is IT" ; one of the order 11, this is U^H ; 
one of the order 9, this is I7*4> ; one of the order 7, this is UH*\ one of the order 5, 
this is H^ ; and one of the order 3, this is V CT. The line (6) shows that there are 8 
asyzygetic covariants of the degree 6, viz. one of the order 18, this is U^\ one of the 

inyariant or leading coefficient, we have the equations of connexion of these nomerioal coefficients. Thns, for 
the dlBcriminant of a cnhic, the terms of the next inferior weight are ahd^ ab^d, abc\ &*e, and operating on 
each of these separately with the symhol 



ind. 6 . - + 2 ind. c . ■=• + 3 ind. d . - , 
a b c 



we find 



abed 




+ 6a«d' 


3bH 


+ 2abed 




2 6»c« 


+ 6ac« 


■\-Sabcd 




+ 4fc»c> 


+ 8 6»d 



and equating the horizontal lines to zero, and assuming a'd':=l, we have a^=l, abed= -6, ac^=4, b*d=if 
6M= - 3, or the value of the discriminant is that given in the text. 



262 



A SECX)ND MEMOIR UPON QU AN TICS. 



[141 



order 14, this is U*H \ one of the order 12, this is 17** ; one of the order 10, thi^ is 
U^H^\ one of the order 8, this is UH^\ two of the order 6 (Le. the three covariants 
ff', ^ and Vf/"* are not asyzygetic, but are connected by a single linear equation or 
syzygy), and one of the order 2, this is VH. We are thus led to the irreducible 
covariants U, JT, ^, V connected by a linear equation or syzygy between H*, ^' and 
VfT', and this is in fact the complete system of irreducible covariants; V is therefore 
the only invariant. 

36. The asyzygetic covariants are of the form U^H^V^^ or else of the form 
U^H^V^^; and since U, H, V are of the degrees 1, 2, 4 respectively, and ^ is of the 
degree 3, the number of asyzygetic covariants of the degree m of the first form is 
equal to the coefficient of a^ in 1 -r- (1 — x) (1 — a^) (1 — a^), and the number of the 
asyzygetic covariants of the degree m of the second form is equal to the coefficient 
of w'^ in a^^(l— a:) (1 — ic")(l — a?*). Hence the total number of asyzygetic covariants is 
equal to the coefficient of »"* in (1 + a;*) -^ (1 — x) (1 — x*) (I — a^), or what is the same 
thing, in 



and conversely, if this expression for the number of the asyzygetic covariants of the 
degree m were established independently, it would follow that the irreducible invariants 
were four in number, and of the degrees 1, 2, 3, 4 respectively, but connected by 
an equation of the degree 6. As regards the invariants, eveiy invariant is of the 
form V^, i.e. the number of asyzygetic invariants of the degree m is equal to the 

coefficient of ic*" in , ^ , and conversely, fi-om this expression it would follow that 

there was a single irreducible invariant of the degree 4. 

37. In the case of a quartic, the function to be developed is: 

1 

(1 - £r) (1 - arz) (1 - a:»z) (1 - a^^) (1 - a:*^) ' 

and the coefficients are given by the table. 

(0) 

0) 

(2) 
(3) 
(4) 
(5) 
(6) 























1 




1 


1 


1 




1 


1 


2 


2 


3 




1 


1 


2 


3 


4 


4 


5 






1 


1 


2 


3 


5 


5 


7 


7 


8 




1 


1 


2 


3 


5 


6 


8 


9 


11 


11 


12 


1 


1 


2 


3 


5 


6 


9 


10 


13 


14 


16 


16 


18 



141] 



A SECOND MEMOIR UPON QUANTICS. 



263 



and subtracting from each coefficient the coefficient immediately preceding it, we have 
the table : 



























1 


(0) 
















1 








(1) 














1 





1 





1 


(2) 










1 





1 1 

1 


1 





1 
1 


(3) 






1 





1 


t 
1 


2 





2 





(4) 




1 





1 


1 
1 


2 
3 


1 

1 


2 


1 


2 





1 


(5) 


1 





1 


1 


2 


3 


1 


2 





2 


(6) 



the examination of which will show that we have for the quartic the following 
irreducible covariants, viz. the quartic itself U; an invariant of the degree 2, which I 
represent by / ; a covariant of the order 4 and of the degree 2, which I represent by H ; 
an invariant of the degree 3, which I represent by J; and a covariant of the order 6 
and the degree 3, which I represent by <I> ; but that the irreducible covariants are 
connected by an equation of the degree 6, viz. there is a linear equation or syzygy 
between <!>», PH^, PJH^U, IJ^HU^ and J^U^\ this is in fact the complete system of 
the irreducible covariants of the quartic: the only irreducible invariants are the 
invariants /, J. 

38. The asyzygetic covariants are of the form U^I'^H^J*, or eke of the form 
UpI'^H^'J*^, and the number of the asyzygetic covariants of the degree m is equal to 
the coefficient of a^ in (1 + ir')^(l — a:)(l — a:^)*(l — a^), or what is the same thing, in 

1-A^ 



(i-j')(i'-a^y(i-x'y' 



and the asyzygetic invariants are of the form /pJ^, and the number of the asyzygetic 
invariants of the degree m is equal to the coefficient of af^ in 1-t-(1 — a;*)(l — a^). 
Conversely, if these formulae were established, the preceding results as to the form 
of the system of the irreducible covariants or of the irreducible invariants, would 
follow. 

39. In the cas^ of a quintic, the function to be developed is 



(I " z)(l - xz){l - a^z)(l ^ .z^z) (I -a^z)(l-a^z)' 



and the coefficients are given by the table : 



264 



A SECOND MEMOIR UPON QUANTICS. 



[1 















1 






1 


1 


1 








1 


1 


2 


2 


3 


3 






1 


1 


2 


3 


4 


5 


6 


6 






1 


1 


2 


3 5 


6 


8 


9 


11 


11 


12 


1 


1 


2 


3 


5 


7 


9 


11 


U 


16 


18 


19 


20 





















































(0) 

(1) 

(2) 
(3) 
(4) 
(5) 



and subtracting from each coefficient the one which immediately precedes it, we h 
the table : 



1 







1 







1 



1 



1 







1 



1 



1 







1 



1 



1 







1 



(0) 

0) 
(2) 
(3) 
(4) 
(5) 



We thus obtain the following irreducible covariants, viz. : 

Of the degree 1 ; a single covariant of the order 5, this is the quintic itself 

Of the degree 2 ; two covariants, viz. one of the order 6, and one of the order 2 

Of the degree 3 ; three covariants, viz. one of the order 9, one of the order 5, i 
one of the order 3. 

Of the degree 4 ; three covariants, viz. one of the order 6, one of the order 4, i 
one of the order (an invariant). 

Of the degree 6 ; three covariants, viz. one of the order 7, one of the order 8, i 
one of the order 1 (a linear covariant). 

These covariants are connected by a single syzygy of the degree 5 and of 
order 11 ; in fact, the table shows that there are only two asyzygetic covariants 
this degree and order; but we may, with the above-mentioned irreducible covaria 
of the degrees, 1, 2, 3 and 4, form three covariants of the degree 5 and the or 
11 ; there is therefore a syzygy of this degree and order. 



141] A SECOND MEMOIR UPON QU ANTICS. 265 

40. I represent the number of ways in which q can be made up as a sum of 
m terms with the elements 0, 1, 2, . . . m, each element being repeatable an indefinite 
number of times by the notation 

P(0, 1, 2, ...m/j, 
and I write for shortness 

P'(0, 1, 2, ...m/? = P(0, 1, 2...m/g-P(0, 1, 2 ... m/(j- 1). 

Then for a quantic of the order m, the number of asyzygetic covariants of the degree 
and of the order /i is 

P'(0, 1, 2...m/i(m5-A*). 

In particular, the number of asyzygetic invariants of the degree 6 is 

P'(0, 1, 2...mf^e. 

To find the total number of the asyzygetic covariants of the degree 6, suppose 
first that md is even ; then, giving to /x the successive values 0, 2, 4, . . . mO, the 
required number is 

P{^e) -PQm^-l) 

+ P(im5-l)~P(Jm^-2) 

+ P(2) -P(l) 

+ P(1) 
= P(W)> 
L e. when md is even, the number of the asyzygetic covariants of the degree 6 is 

P(0, 1, 2...m/imd; 

and similarly, when md is odd, the number of the asyzygetic covariants of the degree 
d is 

P(0, 1, 2, ...m)•i(»^5-l)• 
But the two formulae may be united into a single formula; for when inB is odd \md 
is a fraction, and therefore P{^ff) vanishes, and so when mB is even \{m6—\) is a 
fraction, and Pi(m5 — 1) vanishes; we have thus the theorem, that for a quantic of 
the order m\ 

The number of the asyzygetic covariants of the degree 6 is 

P(0, 1, 2...m)«im5 + P(0, 1, 2, ... m)«Kwid- !)• 

41. The functions P{\m0\ &c. may, by the method explained in my "Researches 
on the Partition of Numbers," [140], be determined as the coefficients of a^ in certain 
functions of a? ; I have calculated the following particular cases : — 

Putting, for shortness, 

P'(0, 1, 2,... m)* ^m5 = coefficient of in <f>m, 
C. II. 34 



266 A SECOND MEMOIR UPON QUANTICS. [141 

then 4>2 = 



then 



^ = 



1 



^ (l-a;»)(l~aj»)' 

Afi - (1 -a? )(l+a?-a:'-a?*-- je'+jg^ + a:») 
'''^ "■ (l-««)«(l-a:»)(l-a;*)(l-a:») ' 

.Q^ (l--a?)(l+a?-d^~a;* + a^ + af + g» + g» + a^<>-g» + a;^» + a:^8) 



(l-a;»)«(l-a:»)«(l~j;*)(l-a:»)(l-a;') 
P(0, 1, 2, ... m)* ^md = coefficient of a:^ in -^m, 



then i/^2 = 



Vr3 = 



l+ar* 
(l~a;»)'(l-a;*)' 



''' (l-a?)«(l-a;»)(l~a^)* 

1 + a:* + 6a:* + 9^* + 12g» + Qx"^ -k- 6a^ + x"^ -^ a^"" 

P(0, 1, 2, ... m)*^md— 1) = coefficient of a:;* in -^m, 

''^ (l-a»)«(l-a;*)' 

''^ (l-a:»)»(l-«*)(l -«•)(! -ic') 

And from what has preceded, it appectrs that for a quantic of the order m, the 
number of asyzygetic covariants of the degree ^ is for m even, coefficient a^ in -^i, 
and for m odd, coefficient a^ in (yp^m + yfr^m); and that the number of asyzygetic 
invariants of the degree is coefficient a:^ in ifym. Attending first to the invariants: 

42. For a quadric, the number of asyzygetic invariants of the degree is 

1 



coefficient of in 



l-a;»' 



which leads to the conclusion that there is a single irreducible invariant of the 
dijgree 2. 



141] A SECOND MEMOIR UPON QUANTICS. 267 

43. For a cubic, the number of asyzygetic invariants of the degree is 

coefficient a? in .j — -., 

1 — ar 

i.e. there is a single irreducible invariant of the degree 4. 

44. For a quartic, the number of asyzygetic invariants of the degree is 

coefficient of in (i _ J^^^ _ ^) . 
i.e. there are two irreducible invariants of the degrees 2 and 3 respectively. 

45. For a quintic, the number of asjrzygetic invariants of the degree is 

l^af + a^ 



coefficient «• in 



(l-a;*)(l-ir«)(l-a^)' 



The numerator is the irreducible factor of 1 — ic", Le. it is equal to (1 — a;") (1 — «•) 
-r(l — a:") (1 — a:") ; and substituting this value, the number becomes 

1 — ^ 
coefficient a:^ in 



(l-a;*)(l-a*)(l-a:")(l-d5»)' 



Le. there are in all four irreducible invariants, which are of the degrees 4, 8, 12 and 
18 respectively; but these are connected by an equation of the degree 36, i.e. the 
square of the invariant of the degree 18 is a rational and integral function of the 
other three invariants; a result, the discovery of which is due to M. Hermite. 

46. For a sextic, the number of asyzygetic invariants of the degree is 

/»• X a - (l'-x)(l'\-x — a^ — a^ — a:^ + w' + a^) 

coefficient ar in ^^ — /■. ^x, /■. ijwi zitti =;\ • 

(1 - a:;*)* (1 - a^) (1 - a?*) (1 - of) 

the second factor of the numerator is the irreducible fiswtor 1 — a:**, i. e. it is equal 
to (l-aJ»)(l-a:»)(l-a^)(l~a:»)4-(l-a:")(l-a^<>)(l-aj»)(l-a;); and substituting this 
value, the number becomes 

l-a:» 



coefficient as* in 



(1 - a:') (1 - a?*) (1 - a;*) (1 - x"') (1 - ai'') ' 



Le. there are in all five irreducible invariants, which are of the degrees 2, 4, 6, 10 

and 15 respectively; but these are connected by an equation of the degree 30, i.e. 

the square of the invariant of the degree 15 is a rational and integral function of 
the other four invarianta 

47. For a septimic, the number of asyzygetic invariants of the degree is 

l-a:«-f2a;»-a^* + 6a^ + 2a^* + 6a:^« + 2a:" + 5a:»-a;«+2a;^-a:« + a* 



coefficient a;* in 



(1 - a?*)(l ~a^)(l -a;«)(l -a;^»)(l - x"') 

34—2 



268 A SECOND MEMOIR UPON QUANTICS. [141 

the numerator is equal to 

(1 - a:-)(l -a:»)-*(l -ir^»)(l -a:")-*(l -«")-^.... 

where the series of factors does not terminate; hence [incorrect, see p. 253] the number 
of irreducible invariants is infinite; substituting the preceding value, the number of 
asyzygetic invariants of the degree is 

coefficient a:^ in (1 - aj*)-> (1 - a^y' (1 " ^^ (1 - ^^ V- • • 

The first four indices give the number of irreducible invariants of the corresponding 
degrees, i.e. there are 1, 3, 6 and 4 irreducible invariants of the degrees 4, 8, 12 and 
14 respectively, but there is no reason to believe that the same thing holds with 
respect to the indices of the subsequent terms. To verify this it is to be remarked, 
that there are 1, 4, 10 and 4 asyzygetic invariants of the degrees in question respect- 
ively; there is therefore one irreducible invariant of the degree 4; calling this Z4, 
there is only one composite invariant of the degree 8, viz. X/; there are therefore 
three irreducible invariants of this degree, say Xg, Xg', Xg". The composite invariants 
of the degree 12 are four in number, viz. X^*, X^X^, X^X^\ X^X^\ and these cannot he 
connected by any syzygy, for if they were so, X^\ Xg, Xg', X," would be connected by a 
syzygy, or there would be less than 3 irreducible invariants of the degree 8. Hence 
there are precisely 6 irreducible invariants of the degree 12. And since the irreducible 
invariants of the degrees 4, 8 and 12 do not give rise to any composite invariant of 
the degree 14, there are precisely 4 irreducible invariants of the degree 14 

48. For an octavic, the number of the asyzygetic invariants of the degree is 



coefficient ar* in 



(1 -a;)(l -f x-a^- a^ + a^ + a?' -h a;* + ic* + a?** - a?*' - «*' -h «" + «*•) . 



{l-x'yil -a:»)«(l -a:*)(l -ar»)(l ^af) 
and the second factor of the numerator is 

(1 -a:)-»(l -a:»)(l -ar')-»(l -a:V(l -^)"' (l-af)-'(l -ar»V(l -a;**)(l -a?*0(l -^') - 

where the series of factors does not terminate, hence [incorrect] the number of irreducible 
invariants is infinite. Substituting the preceding value, the number of the asyzygetic 
invariants of the degree is 

roeffla;* in (1 - j;*)-' (1 - ar»)-i (1 - ar*)-^! - 4^)-> (1 - a:«)-i (1 - ar^-^ 

There is certainly one, and only one irreducible invariant for each of the degrees 
2, 3, 4, 5 and 6 respectively; but the formula does not show the number of the irre- 
ducible invariants of the degrees 7, &c. ; in fact, representing the irreducible inva- 
riants of the degrees 2, 3, 4, 5 and 6 by X„ X,, X4, X5, Xg, these give rise to 3 com- 
posite invariants of the degree 7, viz. X,X^j, XjXj, X,X4, which may or may not be 
connected by a syzygy; if they are not connected by a syzygy, there will be a single 
irreducible invariant of the degree 7 ; but if they are connected by a sjzygy, there 
will be two irreducible invariants of the degree 7 ; it is useless at present to pursue 
the discussion further. 



\ 



141] A SECOND MEMOIR UPON QUANTICS. 269 

Considering next the covariants, — 

49. For a quadric, the number of asyzygetic covariants of the degree is 

1 



coeflScient of in 



(\-x){\-<d^y 



i.e. there are two irreducible covariants of the degrees 1 and 2 respectively; these 
are of course the quadric itself and the invariant. 

50. For a cubic, the number of the asyzygetic covariants of the degree 6 is 

coefficient of m ^^_Jy^^^^) • 

The first £Etctor of the numerator is the irreducible factor of 

l-a^, = (l-a^)-(l-fl?), 
snd the second factor of the numerator is the irreducible factor of 

l-aj*, = (l-aj*)^(l-a^); 
^substituting these values, the number is 



coefficient of in 



(1 -a?)(l -a;»)(l -a:»)(l -iT*)' 



i.e. there are 4 irreducible covariants of the degrees 1, 2, 3, 4 respectively; but these 
^re connected by an equation of the degree 6; the covariant of the degree 1 is the 
oubic itself U, the other covariants are the covariants already spoken of and repre- 
sented by the letters JJ, 4> and V respectively {H is of the degree 2 and the order 3, 
<X> of the degree 3 and the order 3, and V is of the degree 4 and the order 0, 
x.e. it is an invariant). 

51. For a quartic, the number of the asyzygetic covariants of the degree 6 is 



coefficient sfi in 



(l-«)«(l-aj»)(l-aj»)* 



^he numerator of which is the irreducible factor of l—af^ i.a it is equal to 
(1 — «•) (1 — «) -r (1 — oj*) (1 — aj*). Making this substitution, the number is 

1 — ir* 
coefficient of in 



(l-a?)(l-a^)»(l-a^)«' 



X.e. there are five irreducible covariants, one of the degree 1, two of the degree 2, 
^,nd two of the degree 3, but these are connected by an equation of the degree 6. 
*irhe irreducible covariant of the degree 1 is of course the quartic itself U, the other 
irreducible covariants are those already spoken of and represented by /, H, J, 4> 
^"espectively (/ is of the degree 2 and the order 0, and J is of the degree 3 and 
^he order 0, Le. / and J are invariants, H is of the degree 2 and the order 4, 4> 
xs of the degree 3 and the order 6). 



270 A SECOND MEMOIR UPON QUANTICS. [141 

52. For a quintic, the number of irreducible covariants of the degree is 

the numerator of which is 

(l+a?)»(l-a?H-2a;»H-ir«H-2a^H-S«» + ««H-5a;'H-a^H-8a;* + 2a^'H-a?" + 2a;"-a^ 
the first £Eu;tor is (1 — ai)~^ (1 — a^y, the second factor is 

(1 -^)(1 -a^)-«(l -aj»)-»(l -a:*)-^(l -af)'^(l -afiy{l-x'y(l''a*y(l^a^y(l--x'^)-^(l^a!''y^..., 

which does not terminate ; hence [incorrect] the number of irreducible oovariants is 
infinite. Substituting the preceding values, the expression for the number of the 
asyzygetic covariants of the degree 6 is 

coeff. ^in (l-x)-^ (1-a'«)-«(1 -a;»)-«(l -.d?«)-»(l -a?*)-«(l -«•)*(! - x^)^ (I - a*f (I -j^)i(l -ar»«)-«(l -^»)"**.... 

which agrees with a previous result: the numbers of irreducible covariants for the 
degrees 1, 2, 3, 4 are 1, 2, 3 and 3 respectively, and for the degree 5, the number 
of irreducible covariants is three, but there is one syzygy between the composite 
covariants of the degree in question ; the difference 3 — 1 » 2 is the index taken with 
its sign reversed of the factor (1 — aj")~*. 

53. I consider a system of the asyzygetic covariants of any particular degree and 
order of a given quantic, the system may of course be replaced by a system the terms 
of which are any linear functions of those of the original system, and it is necessary 
to inquire what covariants ought to be selected as most proper to represent the 
system of asyzygetic covariants; the following considerations seem to me to furnish 
a convenient rule of selection. Let the literal parts of the terms which enter into 
the coeflScients of the highest power of x or leading coefficients be represented by 
Ma, Mp, My,,., these quantities being arranged in the natural or alphabetical order; 
the first in order of these quantities M, which enters into the leading coefficient of a 
particular covariant, may for shortness be called the leading term of such covariant, 
and a covariant the leading term of which is posterior in order to the leading terra 
of another covariant, may be said to have a lower leading term. 

It is clear, that by properly determining the multipliers of the linear functions we 
may form a covariant the leading term of which is lower than the leading term of 
any other covariant (the definition implies that there is but one such covariant); call 
this 0. We may in like manner form^ a covariant such that its leading term is lower 
than the leading term of every other covariant except 0i ; or rather we may form a 
system of such covariants, since if 4>, be a covariant having the property in question, 
<l)j + Ar0i will have the same property, but k may be determined so that the covariant 
shall not contain the leading term of 6i, i.e. we may form a covariant B, such that 
its leading term is lower than the leading term of every other covariant excepting 
01, and that the leading term of ©i does not enter into 0,; and there is but one such 
covariant, 0,. Again, we may form a covariant 0, such that its leading term is lower 
than the leading term of every other covariant excepting 0i and 0,, and that the 



141] 



A SECX)ND MEMOIR UPON QUANTIC8. 



271 



leading terms of Bi and B, do not either of them enter into B, ; and there is but 
one such co variant, B,. And so on, until we arrive at a covariant the leading term 
of which is higher than the leading terms of the other covariants, and which does 
not contain the leading terms of the other covariants. We have thus a series of 
covariants Si, B„ B,, &c. containing the proper number of terms, and which covariants 
may be taken to represent the asyzygetic covariants of the degree and order in question. 

In order to render the covariants B definite as well numerically as in regard to 
sign, we may suppose that the covariant is in its least terms (Le. we may reject 
numerical £Etctors common to all the terms), and we may make the leading term 
positive. The leading term with the proper numerical coeflScient (if different from 
unity) and with the proper power of x (or the order of the function) annexed, will, 
when the covariants of a quantic are tabulated, be sufficient to indicate, without any 
ambiguity whatever, the particular covariant referred to. I subjoin a table of the 
covariants of a quadric, a cubic and a quartic, and of the covariants of the degrees 
1, 2, 3, 4 and 5 respectively of a quintic, and also two other invariants of a quintic. 

[Except for the quantic itself, the algebraical sum of the numerical coefficients 
in any column is =0, viz. the sum of the coefficients with the sign + is equal to 
that of the coefficients with the sign — , and I have as a numerical verification 
inserted at the foot of each column this sum with the sign ±]. 



( 



Covariant Tables (Noa 1 to 26). 
No. 1. No. 2. 

5 ^» yY' 



a+l 


6 + 2 


c + 1 




=fc 1 



The tables Noa 1 and 2 are the covariants of a binary qi^adric. No. 1 is the 
quadric itself; No. 2 is the quadrin variant, which is also the discriminant. 

No. 3. No. 4. 




H^. UY 




\^y i/Y 



± 1 



± 1 



± 1 



No. o. 



No. G. 



a-c/-f 1 


a6rf+3 


acrf-3 


OiP -1 


a6c~ 3 


ac" -6 


b*d +6 


bed + 3 


A» +2 


b^c +3 


bc^ -3 


c» -2 



5-, yr 



:L H 



:i^6 



± 6 



± 3 



a'd' 


+ 1 


abed 


-6 


a^ 


+ 4 


bH 


+ 4 


6»c« 


- 3 



± y 



The tables Noa 3, 4, 5 and 6 are the covariants of a binary cubic. No. 3 is the 
cubic itself; No. 4 is the quadricovariant, or Hessian; No. 5 is the cubicovariant ; 
No. 6 is the invariant, or discriminant. And if we write No. 3 = IT, No. 4 = if. 
No. 5 = *, No. 6 = V, 



272 



A SECOND MEMOIR UPON QUANTICS. 



then identically, 



No. 7. 



a+1 


6 + 4 


C4- 6 


(^ + 4 


« + l 



$«.y)* 



No. 


8. 


00 + 


1 


W- 


4 


c« + 


3 



No. 9. 



( 



±* 



oe + l 
6«^1 


ad + 2 
6c-2 


06+ 1 

6rf+2 
c«-3 


6« +2 
crf-2 


6J+1 
c« -1 



$^y)* 



Jr 1 



±2 



±3 



db2 



± 1 



No. 


10. 


ac6 


+ 1 


OflP 


-1 


b^e 


-1 


bed 


+ 2 


c» 


- 1 



No. 11. 



±s 



o«rf+l 


o*« + 1 


o56 + 5 


oe« 


a(f0<- 5 


a^ -1 


6tf« - 1 


o6c-3 


o6i+2 


aed^ 15 


o^-lO 


6c0 + 15 


6<fe-.2 


C6fe -f 3 


6» +2 


oc«-9 


6y + 10 


6»tf + 10 


bd'-lO 


c"tf + 9 


rf» -2 




6»c +6 


6c* 


bed 

er • . • 


<?d 


crf»-6 





^ 



^8 



:1:9 



:k 15 



dblO 



:l:16 



:t9 



±8 



No. 


12. 


oV 


+ 


1 


o«6c£e» 


— 


12 


oVe* 


— 


18 


a^cd^e 


-f 


54 


a^d^ 


— 


27 


ab'e^ 


+ 


54 


ab'd'e 


— 


6 


abc^de 


— 


180 


abecP 


+ 


108 


oc^ 


+ 


81 


aA?d} 


— i 


54 


6V 


— 


27 


6'o<^ 


+ 


108 


6»ci» 


— 


64 


6V« 


— 


54 


6V(i* 


+ 


36 


6c*rf 




• • • 


c« 




• • • 



^442 

The tables Nos. 7, 8, 9, 10 and 11 are the irreducible covariants of a qu 
No. 7 is the quartic itself; No. 8 is the quadrinvariant ; No. 9 is the quadricova] 
or Hessian; No. 10 is the cubinvariant ; and No. 11 is the cubicovariant. The 
No. 12 is the discriminant. And if we write No. 7 = CT, No. 8 = /, No. 9 
No. 10 = J, No. 11 = *, No. 12 = V, 

then the irreducible covariants are connected by the identical equation 

jir»-/f7«fr+4fr» + 4)« = o, 

and we have 

V = /»-27J«. 



141] 



A SECOND MEMOIR UPON QUANTICS. 



273 



[The Tables Nos. 13 to 24 which follow, and also Nos. 25 and 26 which are given 
in 143 relate to the binary quintic. I have inserted in the headings the capital letters 
A, B, . . . L and also Q and Q' by which I refer to these covaiiants of the quintic. A is 
the quintic itself, C is the Hessian, G is the quartin variant, J a linear covariant: Q is 
the simplest octinvariant, and Q' is the discriminant. As noticed in the original memoir 
we have AI + BF-CE = 0; and Q' = G« - 128 Q, only the coefficient 128 was by mistake 
given as 1152.] 

A. No. 13. 



( 



a + 1 


6 + 5 


c+10 


rf+10 


6 + 5 


/+i 



Xa^y)' 



B. No. 14. 



( 



ae+1 
bd-i 
c* +3 


a/+l 
6e-3 
ed+ 2 


kf+1 
ce-4: 

d»+3 



tn^ y)* 



:1:4 :1:8 ±4 

C. No. 15. 



( 



oc + 1 

6* -1 


ad+3 
be -3 


ae + 3 

W+3 
c" -6 


(l/'+l 

6« +7 
crf-8 


6/ + 3 
cd +3 
cP -6 


C/ + 3 
de-Z 


d/^l 
e» -1 



H^ yY 



db 1 



=k8 



:t 6 ^S ±6 

D. No. 16. 



^B 



=fc 1 



( 



a^ ... 


06/"+ 1 


a4f*\ 


o«^ ... 


ace + 1 


ode— 1 


w? -1 


6<ir+i 


atP-1 


jy-i 


6c^-l 


W -1 


Va -1 


&C0 + 1 


5<2«-i-l 


«y-i 


hcd+2 


M»+l 


«»« +1 


ed0 + 2 


c* -1 


c'rf-l 


aP-l 


<i» -1 



$«, y)' 



:^8 



8 



8 



8 







K 


No. 17. 






ay+1 


ay+ 5 


acf-^ 2 


adf- 2 


cw/- 5 


qr- 1 


a6d - 5 


ace - 16 


ocie-12 


oa* - 8 


64/'+ 16 


6i2/-+5 


, 0(^+2 
V 6*^ + 8 


a<i'+ 6 


6!/'+ 8 


6c/ +12 


6«« + 9 


c(i/'-2 


6»6 - 9 


6c« -38 


6(20+38 


c*/- 6 


C6* -8 


he -6 


5c(2+38 


M> + 72 


i?e -72 


ce20-38 


(^6 +6 




c» -24 


c'rf-32 


c«P + 32 


d» +24 




:1:11 


:t49 


:1:S2 


:1>S2 


:i:49 


dbii 



$^. yy 



F. No. 18. 



I 



(i»rf+l 
a6c-3 
6» +2 



aV + 



2 

a6<;+ 1 
oa" -12 
6»c + 9 



a^f^ 1 
a6a + ll 
occf- 34 
6»rf+16 
6c» + 6 



a6/+ 7 
ace — 8 
arf«-34 
6*6 +29 
bed- 2 
c* + 8 



a^+ 5 
acfo-40 
6*/ + 16 
6c6 + 47 
6c^-44 
ed + 16 



o^/-- 5 
ac« -16 
6c/ + 40 
bde-^7 
&e +44 
c(P -16 



oc/- 7 
6(f/+ 8 
6«» -29 
c«/ + 34 
e£f0+ 2 
d» - 8 



<!/••- 1 
6e/-ll 
«i^+34 
(»» -16 
cPtf - 6 



6/«- 2 
<»/- 1 
rf»/+12 
(fo« - 9 



C/^-1 
cis/'+S 
e» -2 



:l:8 :& 12 

C. II. 



:k84 



:1:44 



:k84 



84 



^44 



:S=84 



^\2 



35 



$«^y)' 



274 



A SECOND MEMOm UPON QUANTICS. 



[141 



G. No. 19. 

«!/•*+ 1 

a6(?/- 10 
acdf-^ 4 
occ* + 16 
ad^e- 12 
hHf^ 16 
6V + 9 
6cy -12 
hcde - 76 
hd" +48 
c^tf + 48 
c^d^ - 32 



H. No. 20. 



^ 142 



a«4/'+ 1 


aV + 2 


ay « + 1 


a6/«+ 3 


a^«+ 1 


a V 


a6^- 4 


a6«/ - 4 


acef- 4 


adef- 3 


aftc/- 3 


aM -10 


occj/'- 2 


aciy- 2 


a«« + 2 


aftc2s - 5 


ac»/- 2 


ace" + 4 


oc^ + 4 


6y« ... 


ao*« + 10 


oecfe + 24 


a^i'e 


6V - 10 


beef - 5 


/ acd^ — 4 


0(^-12 


ft«rf/+ 4 


6ai^+ 24 


bd"/-^ 10 


^ 6'/ + 2 


6»</ + 4 


6V - 9 


ftc«» + 16 


M«» - 5 


h^ce - 5 


W« +16 


W/ ... 


6c^e -22 


c»4/-- 4 


6«<^ +14 


ftc«e -22 


6ccfo + 50 


cy -12 


c*fi" + 14 


he'd - 16 


6cc^ - 4 


W -36 


c«<fo - 4 


crf«« - 16 


c* + 6 

i 


c«rf + 8 


c»« -36 
cW +28 


cc^ + 8 


rf* + 6 


dbSS 


db54 


j=87 


db54 


=k83 



5*. i^)* 



I. No. 21. 



ttV* + 1 


a«ei/+ 2 


a*^ 


ay« ... 


abp .., 


ac/«- 2 


«(/••- 1 


a*(ie - 1 


aV - 2 


abd/ + 2 


cU>e/ 


ac«/-2 


culef 


a^y +1 


' a6y-l 


a^/-10 


a6e" -2 


acdf ... 


acV+l 


ae> + 2 


her + 1 


abce - 2 


a6c^+ 10 


ocy-l 


a<j«» -20 


arf(5» + 1 


6y» + 2 


6cfe/+2 


a6«/* + 4 


ac** 


acde-2 


arf*« + 20 


6y +2 


beef 
bSf-^ 10 


6«* -3 


. ac*rf - 1 


acd^ 


cuP +S 


b^d/ + 20 


M^+2 


c V - 4 


< 6»e +3 


6y - 2 


b'cf - 1 


6V 


6<w* -5 


6cfe« -14 


«P/ + 1 


b^cd -6 


6»c« + 14 


b^de +6 


W/ -20 


bd'e - 1 


c«(y-10 


ocfe* +6 


ftc* +3 


b^d' + 2 


6c»«5 +1 


&C(i0 


cy -3 


c*«» - 2 


(i»« -3 




W(/ -26 


6a^ -9 


6rf» -20 


c»ci(5 +9 


ccTe + 26 




1 


c* + 12 


c*rf +4 


c»e +20 


cd" -4 


dl" -12 




i 11 


=k40 


dbi5 


=b0O 


J. 15 


db40 


dbll 



5«, y)'- 



J. No. 22. 



o»c/« + 1 


a»4/"»+ 1 


a'def- 2 


aV/ - 1 


aV + 1 


o6c/»- 2 


ab*/*- 1 


abdef- 4 


aJc?/"- 4 


o&e' + 6 


aM'/+ 8 


ac"*/" + 8 


tM^- a 


aaPf- 2 


ocM^- 2 


a«fe'-12 


w»f + 14 


ad*e + 6 


. acd»« - 22 
< ad* + 9 


*»/• + 1 


6»«/ - 2 


6V + 6 


6V/ + 14 


Vecif-\2 


VdU? -16 


6'c«« -16 


fte»4r-22 


iWj +10 


6«V +10 


ftc*/ + 6 


iedV +30 


6e'<2« +30 


W -16 


6c«i» -20 


c*/ + 9 


«rt» -16 


«:•«*» -20 


c»d' +10 


«•«<» +10 



\^yY 



:1:95 



db95 



141] 



A SECOND MBMOIB UPON QUANTICS. 



275 





K 


No. 23. 




aV* ... 


a\p + 1 


a*e(/^- 1 


aV' •• 


c^cef + 1 


a*e^/- 5 


aV/ + 1 


aMf-- 1 


a*cy- 3 


aV + 4 


€^P + 5 


aW/+ 1 


a»^ + 2 


a^/«- 1 


cMef-- 8 


acy«+ 3 


o^V- 1 


ahcef + 8 


a6e» + 3 


(Kidef" 14 


€^cdfAr 14 


a6i«/+ 11 


ocV-H 


ace* + 8 


a^e» - 11 


aW6«- 17 


ocrfy+ll 


ad^f + 9 


oftrf"* - 1 


ac«(^- 11 


(kcd^ + 6 


ewTe* - 6 


ocy - 9 


ac»e» -16 


oc^e - 6 


W^ " 2 


. ac*flfe 4- 14 
V occP - 6 


cuid^e + 44 


6»/» - 4 


h'def + 11 


oc^* - 18 


ft*ce/ 4- 17 


6V - 9 


h^df - 8 


ft»6/ - 3 


5«rf"/ + 16 


hc'ef + 1 


ft»(5» + 9 


h^cdf ^ 6 


h^d^ - 21 


fccrfy - 14 


ft*cy + 6 


6«c«> +21 


6c»e(/'-44 


hcde" + 16 


6^cei« - 16 


6We - 6 


W(5» + 5 


5rf»« - 3 


6»rf» + 8 


6cy + 6 


hcd^t + 39 


c*^/' + 6 


6c»d + 3 


h<?dt - 39 


5^/* - 12 


C»6» - 8 


hc'iP - 2 


6crf» + 22 


cy + 18 


c»cP« + 2 


<?*(/ 


cV + 12 


c»rftf - 22 


cc?* 




c»rf» - 8 


c«(^ + 8 





5^, yY 



±57 



±139 



±129 



±67 



L. No. 24. 



( 



a^hdf ... 

aV/ + 2 
a'cc2e — 5 
aV +3 
ai^cf --^ 
at^de + 6 
aJlx^e + 5 
abed:'''! 
a^d + 1 
6*/ +2 
6»c<!J -5 
ft»(f« -2 
6 Vrf + 8 
6c* -3 



«•/• ... 
a*6^ ... 
c^cdy-k- 7 
a^ce" - 10 
aWtf+ 3 
a6^- 7 
a^«» + 10 
oftcy- 7 

O^CG^tf— 8 

rtW» + 9 
oc^tf +22 
ac«rf»- 19 
ft*c/ + 7 
6»dd + 2 
W« - 19 

bi^d +33 
c» -12 



aV' ... 
a\?«/"- 3 

(^dy+12 

a«<fo«- 9 

a6V+ 3 
a6c^-18 

a6c6"- 18 

aW*e + 30 

w?f - Z 

00*6^ + 45 

a«i» -39 

hi'df -• 6 

6V +27 

6*cy + 16 

6»afo - 87 

6^(^ + 6 

Wtf +12 

6c>(i« + 57 

c« -24 



a*c/»- 1 
a^def-h 7 
aV - 6 
aiy»+ 1 
abce/-26 
aWy+32 
a6<fo«- 8 
ac«4/*- 18 
ac«e* + 6 
euxPe-^62 
ad^ -39 
l^ff +19 
6«c<(/' - 53 
6«ce» +20 
ft'ci'tf -25 
h^f +39 
W<fe - 45 
hcd^ +65 



c*e 



c»(P -20 



a«c(r+ 1 
aV/- 1 
a6^- 7 
abde/-\'26 
cM -19 
oc'e/ - 32 
acdy+18 
aec;^ + 53 
a^e -39 
6»/« + 6 
6»c«/+ 8 
ft*rfy- 6 
6«<^-20 
Mdf^^h 
We» +25 
6crf'tf-52 
6rf* 

c*/ +39 
c»(fo -65 
c>rf» +20 



aV' ... 
a6<(/^+ 3 

oW/- 3 

acy*-12 

ac56/'+18 

occ* + 6 

a^f + 3 

ad'e'-U 

6»^«+ 9 

6»^+18 

6V -27 

bc'ef - 30 

6crfy-45 

6ccfo*+87 



c*cP« 
erf* 



-12 

+ 39 
- 6 
-57 

+ 24 



ay» ... 

a6«/^ ... 
ac4/^- 7 
ace*/ -h 7 
a(Pe/+ 7 
ode" - 7 
6*rf/» + 10 
ft'ey-lO 

bc'r- 3 
6cd;5/'+ 8 
6c<5» - 2 
bd"/ - 22 
6rf»e* + 19 

C»6/ - 9 

c»rf'/+19 
c'de' +11 
cciP« -33 
<f +12 



abp ... 
ac«/*" ... 
arfy-2 
ade^/ + 4 
cw* - 2 

6'(5/* ... 

6c«J/^ + 5 
6cey -5 
bd'ef -5 
We» +5 
cy> -3 

C«rf6/ + 7 

c«e» +2 

crfy -1 

cd»e» -8 
ci*« +3 



5«» y)' 



26 



93 



±207 



241 



±241 



±207 



±93 



±26 






No. 26, Q = + 1 o»cd/' + &C. 
No. 26. 



35—2 



276 



[142 



142. 



NUMERICAL TABLES SUPPLEMENTARY TO SECOND MEMOIR 

ON QUANTICS. 

[Now first published (1889).] 

In the present paper I arrange in a more compendious form and continue to 
a much greater extent the tables (first of each pair) given Nos. 35 — 39 of my 
Second Memoir on Quantics, 141, pp. 260 — 264, which relate to the cubic, the quartic 
and the quintic functions; and I give the like tables for the sextic, the septimic and 
the octavic functions respectively. The cubic table exhibits the coefficients of the several 
xz terms of the function 1-t-(1— ^.1— a?-2r.l— a^^.l— a^z\ or, what is the same thing, 
it gives the number of partitions of a given number into a given number of parts, 
the parts being 0, 1, % 3, (repetitions admissible) : or again, regarding the letters 
a, 6, c, d, as having the weights 0, 1, 2, 3 respectively, it shows the number of literal 
terms of a given degree and given weight. And similarly for the quartic, quintic, sextic, 
septimic and octavic tables respectively, the parts of course being 0, 1,... up to 4, 5, 
6, 7 or 8, and the letters being a, &, ... up to e, /, g^ h or t. The extent of the 
tables is as follows: 

cubic table extends to deg*weight 18 — 27 



quartic 

quintic 

sextic 

septimic 

octavic 



it 



18—36 
18—45 
15—45 
12—42 
10—40 



viz. for the quintic, the sextic and the octavic functions these are the deg-weighta 
of the highest invariants respectively. I designate the Tables as the od-, a^-, a/"-, og-i 
ah' and ai-tables respectively. 

It is to be noticed that in the several tables the lower part of each column is 
for shortness omitted ; the column has to be completed by taking into it the series 



142] NUMERICAL TABLES SUPPLEMENTARY TO SECOND MEMOIR ON QUANTICa 277 

of bottom terms of each of the preceding columns: thus in the af- or quintic table 
the complete column for degree 3 would be 



D 8 




W 8-7 




■V O 


6 


— 1 


6 


- 8 


5 


— 8 


4 


— 4 


3 


■V n 


2 


— 6 


1 


— 7 


1 



where the concluding terms 2, 1, 1 are the bottom terms of the three preceding 
columns respectively. And the meaning is that for degree 3, and weight 8, or 7, the 
number of terms is = 6 ; for weight 7 — 1, =6, the number of terms is =» 6 ; and 
similarly for weights 5, 4, 3, 2, 1, the numbers are 5, 4, 3, 2, 1, 1 ; the numbers are 
those of the terms 



W. 



8 



8 



at 



c?h 



a'e 


a*d 


aU 


«y 


abf 


acf 


adf 


<!&' 


abe 


abd 


ahe 


ace 


ode 


tuf 




6' 


(U? 


acd 


ad? 


hV 


be/ 






6'c 


l?d 


6V 


b*ce 


bde 








he 


bed 


bcP 
<?d 





No. 



8 



The like remarks and explanations apply to the other table& 



D 1 8 


8 


4 


6 


6 


7 


8 


od-TABLE. 

9 10 u 


18 


18 


14 


16 


16 


17 


18 




W 8-18 


6-4 


6 


8-7 


9 


11-10 


18 


14-18 16 


17-16 


18 


80-19 


81 


88-88 


84 


86-86 


87 




1 1 2 

LI. 


3 
3 


5 
4 
4 


6 

6 
5 


8 
8 
7 
7 


10 

10 

9 

8 

1. 


13 

12 
12 
11 
10 


15 18 
15 18 
14 17 
13 17 
12 15 


21 
21 
20 
19 
18 
16 


25 
24 
24 
23 
22 
20 
19 


28 
28 
27 
26 
25 
23 
21 


32 
32 
31 
31 
29 
28 
26 
24 


36 
36 
35 
34 
33 
31 
29 
27 


41 
40 
40 
39 
38 
36 
35 
32 
30 


45 
45 
44 
43 
42 
40 
38 
36 
33 


50 

50 
49 
49 
47 
46 
44 
42 i 
39 i 
37 1 


— 




1 


- 8 






1 


>4 






14 


— 6 






* 


— 6 








— 7 








» 



278 NUMERICAL TABLES SUPPLEMENTARY TO SECOND MEMOIR ON QUANTICS. [142 



D 12 8 4 6 



6 



8 



(w-TABLE. 



10 11 la 18 14 16 



16 17 



18 



W 8 4 6 8 10 12 14 16 18 20 



24 26 



84 86 



113 5 


8 


12 


18 


24 


33 


43 


55 


69 


86 


104 


126 


150 


177 


207 


241 


1 1 2 4 


7 


11 


16 


23 


31 


41 


53 


67 


83 


102 


123 


147 


174 


204 


237 


~1 2 4 


7 


11 


16 


23 


31 


41 


53 


67 


83 


102 


123 


147 


174 


204 


237 


3 


5 


9 


14 


20 


28 


38 


49 


63 


79 


97 


118 


142 


168 


198 


231 




5 


8 


13 


19 


27 


36 


48 


61 


77 


95 


116 


139 


166 


195 


228 






6 


10 


16 


23 


32 


43 


56 


71 


89 


109 


132 


158 


187 


219 








9 


14 


21 


30 


40 


53 


68 


85 


105 


128 


153 


182 


214 










11 


17 


25 


35 


47 


61 


78 


97 


119 


144 


172 


203 










[ 


15 


22 


32 


43 


57 


73 


92 


113 


138 


165 


196 














18 


26 


37 


50 


65 


83 


104 


127 


154 


184 
















23 


33 


45 


60 


77 


97 


120 


146 


175 


















27 


38 


52 


68 


87 


109 


134 


162 


















■ 


34 


46 
39 


62 
53 
47 


80 
70 
6S 


101 
90 
82 


125 
113 
104 


153 
139 
129 


































1 54 


71 
64 


92 
83 
72 


116 

106 

93 











-0 

-1 

-2 

-8 

-4 

-6 

-6 

-7 

-8 

-9 

-10 

-U 

-12 

-U 

-14 

-16 

-16 

-17 

-U 



a/-TABLE. 



D 
W 



1 



8 



8 



10 



11 



12 



13 



14 16 



16 



17 



18 



8-2 6 8-7 10 18-12 16 16-17 20 28- 



26 28-27 80 88-82 86 88-87 40 43-42 46 



1 1 


3 


6 


12 


20 


32 


1 3 


6 


11 


19 


32 




2 


5 


11 


18 


30 




2 


4 


9 


16 


29 




3 


8 


14 


25 






6 


11 


23 






5 


9 

1 7 


19 
16 
12 












10 



49 
48 
46 
43 
39 
35 
30 
26 
21 
17 
13 



73 
71 
70 
66 
63 
57 
52 
45 
40 
33 
28 
22 
18 



102 141 

101 141 

98 137 

93 134 

88 127 

81 121 

74 111 

66 103 



58 
50 
43 
35 
29 
23 



92 
83 
72 
63 
53 
45 
36 
30 



190 

188 

184 

178 

170 

161 

150 

139 

126 

114 

101 

89 

77 

66 

55 

46 

37 



252 
249 
247 
240 
233 
222 
212 
197 
184 
168 
154 
137 
123 
107 
94 
80 
68 
56 
47 



325 
322 
317 
309 
299 
286 
272 
256 
239 
220 
202 
183 
165 
146 
129 
112 
97 
82 
69 
57 



414 
414 
408 
402 
390 
379 
362 
346 
325 
306 
283 
262 
238 
217 
194 
174 
152 
134 
115 
99 
83 
70 



521 
518 
511 
501 
488 
472 
453 
433 
409 
385 
359 
333 
306 
280 
253 
228 
203 
180 
157 
137 
117 
100 
84 



649 
645 
641 
630 
619 
601 
583 
559 
536 
507 
480 
448 
419 
386 
356 
324 
295 
264 
237 
209 
185 
160 
139 
118 
101 



795 
791 
783 
770 
754 
734 
711 
684 
655 
623 
590 
554 
519 
482 
446 
409 
374 
339 
306 
273 
243 
214 
188 
162 
140 
119 



967 
966 
957 
948 
930 
912 
886 
860 
827 
795 
756 
719 
677 
637 
593 
553 
509 
469 
427 
389 
350 
315 
279 
248 
217 
190 
163 
141 



142] NTJMBBICAL TABLES SUPPLEMENTABY TO SECOND MEMOIR ON QU ANTICS. 279 



agf-TABLK 



D 
W 



18 8 



8 



10 



11 



la 



18 



14 



16 






8 


6 


9 


la 


16 


18 


SI 


84 


8T 


80 


88 


88 


89 


43 


46 


1 1 4 


8 


18 


32 


58 


94 


151 


227 


338 


480 


676 


920 


1242 


1636 




1 3 


8 


16 


32 


55 


94 


147 


227 


332 


480 


668 


920 


1232 


1635 




1 3 


7 


16 


30 


55 


90 


146 


221 


330 


471 


664 


907 


1226 


1617 




3 


7 


14 


29 


51 


88 


139 


217 


319 


464 


648 


896 


1203 


1601 




2 


5 


13 


25 


48 


81 


134 


205 


310 


446 


634 


870 


1182 


1565 




4 


10 


23 


42 


76 


123 


196 


293 


431 


608 


847 


1145 


1533 




3 


9 


19 


39 


68 


116 


182 


280 


408 


587 


813 


1113 


1483 






6 


16 


32 


61 


103 


169 


258 


387 


553 


780 


1064 


1435 






5 


12 


28 


52 


94 


152 


241 


359 


525 


737 


1021 


1373 








10 


22 


46 


81 


139 


218 


335 


488 


699 


965 


1316 








7 


18 


37 


71 


121 


199 


304 


455 


650 


914 


1244 










13 


31 


59 


107 


175 


278 


415 


607 


852 


1178 










11 


24 


51 


91 


157 


248 


382 


557 


798 


1102 












19 


40 


78 


134 


222 


341 


512 


733 


1031 












14 


33 
25 


64 
54 


117 
98 


193 
170 


308 
271 


462 
419 


677 
614 


952 

882 


















20 


42 

[ 34 


83 
67 


144 
124 


240 
206 


371 
331 


559 
499 


803 
734 




















26 


56 
43 


103 

86 


180 
151 


289 
253 


449 
394 


661 
596 




( 


















35 


69 
57 


129 
106 


216 
187 


349 
302 


529 
472 
























44 


88 
70 


156 
132 


263 
223 


412 
362 


























58 


108 
89 


192 
159 


303 
270 




























71 


134 
109 


228 
195 






























90 


161 
135 




































110 





1 

3 

8 

4 

6 

6 

7 

8 

9 

10 

11 

18 

18 

14 

16 

16 

17 

18 

19 

80 

81 



84 

86 



-87 



39 

80 



280 NUMERICAL TABLES SUPPLEMENTARY TO SECOND MEMOIR ON QUANTICS. [l42 



oA-TABLE. 



D 
W 



1 8 



8 



10 



u 



la 






4^ 


7 11-10 


14 


18-17 


n 


86-M 


28 


88-81 


86 


88-88 


48 


1 1 4 


10 


24 


49 


94 


169 


289 


468 


734 


1117 


1656 




1 4 


10 


23 


48 


94 


166 


285 


464 


734 


1109 


1646 




1 3 


9 


23 


46 


90 


162 


282 


454 


722 


1093 


1634 


' 


3 


8 


20 


43 


88 


155 


272 


441 


709 


1069 


1605 




2 


7 


19 


39 


81 


146 


263 


424 


686 


1038 


1572 




2 


5 


16 


35 


76 


136 


247 


403 


663 


1000 


1524 




4 


14 


30 


68 


125 


233 


379 


629 


957 


1475 




3 


11 


26 


61 


112 


214 


354 


598 


908 


1410 






9 


21 


52 


100 


197 


325 


558 


856 


1346 






6 


17 


46 


87 


176 


297 


520 


799 


1271 






5 


13 


37 


75 


158 


268 


477 


742 


1197 








10 


31 


63 


137 


239 


437 


682 


1114 








7 


24 


53 


120 


210 


392 


623 


1036 










19 


42 


101 


184 


353 


563 


950 










14 


34 


86 


157 


311 


506 


871 










11 


26 
20 


70 
58 


134 
112 


274 
236 


449 
397 


788 
711 
















15 


45 
36 


93 
75 


204 
171 


346 
300 


633 
564 


















27 


61 


145 


256 


493 














21 


47 
37 


119 
98 


218 
182 


432 
372 




















28 


78 
63 


152 
124 


320 
270 






















48 


101 


229 


















38 


80 
64 


189 
157 


« 






















49 


127 
103 


























81 


























65 





11 

18 
18 
14 
16 
18 
17 
18 
19 



-81 



-84 

-86 
-88 

-87 



89 

80 



NUMERICAL TABLES SUPPLBBiENTABY TO SECOND MEMOIB ON QUANTICS. 281 



W 













ai-TABLE. 













1 8 


8 


4 


6 


6 


7 


8 


9 


10 







4 8 


18 


16 


80 


84 


88 


38 


86 


40 




! . 


1 5 
1 4 


13 
12 


33 
31 


73 

71 


151 

147 


289 
285 


526 
519 


910 
902 


1514 
1502 


— 




— 1 




1 4 


12 


31 


70 


146 


282 


515 


894 


1492 


— 8 




1 3 


11 


28 


66 


139 


272 


499 


873 


1460 


— 8 






3 


10 


27 


63 


134 


263 


486 


851 


1430 


— 4 






2 


8 


23 


57 


123 


247 


461 


816 


1379 


— 6 






2 


7 


21 


52 


116 


233 


440 


783 


1331 


— 6 






5 


17 


45 


103 


214 


409 


738 


1265 


— 7 






4 


15 


40 


94 


197 


383 


696 


1214 


— 8 






3 


11 


33 


81 


176 


348 


645 


1127 


— 9 








9 


28 


71 


158 


319 


597 


1057 


-10 








6 


22 


59 


137 


284 


543 


974 


-U 








5 


18 


51 


120 


255 


495 


900 


-18 










13 


40 


101 


221 


441 


816 


-18 










10 


33 


86 


194 


394 


742 


-14 










7 


25 
20 


70 

58 


164 
141 


345 
302 


662 
593 


-15 






-16 












14 


45 


116 


258 


519 


-17 












11 


36 
27 


97 

77 


222 
185 


457 
393 


-18 






-19 














21 


63 


156 


340 


-80 














15 


48 
38 


127 
104 


286 
243 


-81 






-88 
















28 


82 


200 


-88 
















22 


66 
50 


167 
134 


-84 






-80 


















39 


109 


-86 


















29 


85 
68 


-87 






-88 




















51 


-89 






















40 


-30 



^he numbers of each table are connected in several ways with those of the 
ling tables. One of these connexions, which is of some importance, is best ex- 
d by an example: in the a/*-table, 8-80, the number of terms of degree 8 and 
t 80 is 73 ; and we have 73 = 1 -h 6 + 16 + 23 + 27. viz. (see p. 288) these 
le numbers of the terms in a*, a', a", a\ a* respectively: the complementary 
3, (for example) of a' are be/*, &c. terms in 6, c, d, e, f of the degree 5 and 
t 80, and (replacing therein each letter by that which immediately precedes it) 
are in number equal to the terms in a, 6, c, d, e of the degree o and weight 
= 15 ; thus the number 6 of the terms in question is that for the deg- weight 
f the a6-table: and so 1, 6, 16, 23, 27 are the numbers in the o^-table for 
eg-weights 4-i6, 5-i8, 6-14, 7-i8 and 8-i8 respectively, or (making a change rendered 
lary by the abbreviated form of the tables) say for the deg- weights 4-o, 5-io, 
18 and 8-18. 



•1 



II. 



36 



282 



[143 



143. 



TABLES OF THE CO VARIANTS M TO W OF THE BINARY QUIN 
TIC: FROM THE SECOND, THIRD, FIFTH, EIGHTH, NINTH 
AND TENTH MEMOIRS ON QUANTICS. 



[Arranged in the present form, 1889.] 

The binary quintic has in all (including the quintic itself and the invariants) 
23 covariants, which I have represented by the capital letters, A, B, C, . . . W (alternative 
forms of two of these are denoted by Q' and S'). The covariants A, . . . L, and also 
Q, Q' were given in my Second Memoir on Quantics, and except Q and Q' arc 
reproduced in the present reprint thereof, 141 ; in all these I gave not only the 
literal terms actually presenting themselves, but also the terms with zero coeflBcients; 
in the other covariants however, or in most of them, the terms with zero coefficient*^ 
were omitted. It is very desirable to have in every case the complete series of literal 
terms, and in the covariants as here printed they are accordingly inserted: the number 
of terms is in each case known beforehand by the foregoing q/*-table, 142, and any 
omission is thus precluded; by means of this ci/*- table we have the numbers of terms 
as shown in the following list. 

I have throughout (as was done in the Ninth and Tenth Memoirs) expressed the 
literal terms in a slightly diflferent form from that employed in the Second Memoir: 
this is done in order to show at a glance in each column the set of terms which 
contain a given power of a, and in each such set the terms which contain a given 
power of b. 

The numerical verifications are also given not only for the entire column but for 
each set of terms containing the same power of a; viz. in most cases, but not always, 
the positive and negative coeflScients of a set have equal sums, which are shown by 




43] TABLES OF THE COVABIANTS M TO W OF TH^: BINARY QUINTIC. 283 

number with the sign ± prefixed. The verification is in some cases given in regard 
I the subsets involving the same powers of a and b, here also the sums of the 
)sitive and negative coefficients are not in every case equal. The cases of inequality 
ill be referred to at the end of this paper. 

The whole series of covariants is as follows : 



Hem. 

2 


No. of table 
13 


• 

A 


^ 


(1. 1. 1, 1, 1, IJx, yy 


(leg-weight 
1 (0....5) 


»» 


14 


B 


^ 


(3. 3, 35ar, y)» 


2 (4.6) 


» 


15 





= 


(2, 2. 3. 3. 3, 2, ijx, yf 


2 (2 8) 


11 


16 


D 


ss 


(6. 6, 6, 6$». yy 


3 (6.. 9) 


*i 


17 


E 


= 


(5. 6. 6. 6, 6, 6$«, yy 


3 (5. ...10) 


it 


18 


F 


= 


(3, 4. 5. 6. 6, 6, 6, 5. 4, 3$a:. y)» 


3 (3 12) 


1* 


19 


G 


= 


(12$a;, yy, Invt. 


4-10 


M 


20 


U 


^ 


(11, 11, 12, 11, 11$^, yy 


4 (8., .12) 


)) 


21 


I 


ss 


(9, 11. 11, 12, 11, 11, 9$a!, yy 


4 (7 13) 


»» 


22 


J 


s 


(20, 20$*, yy 


6 (12, 13) 


»» 


23 


E 


= 


(19, 20, 20, 19$«. yy 


5 (11.. 14) 


»» 


24 


L 


= (16, 18, 19, 20, 20, 19, 18, 16$ir, y)' 


5 (9 16) 


8 


83 


M 


= 


(32, 32, 32$«, yy 


6 (14 . 16) 


>i 


84 


N 


= 


(30. 32. 32, 32, 30$a;, y)« 


6 (13.. .17) 


9 


90 





= 


(49, 495a!, yy 


7 (17. 18) 


»> 


91 


P 


= 


(46, 48, 49. 49, 48, 46$*, yy 


7 (15.. ..20) 


2 


Q 25 
Q'26 


Q.Q' 


t 


(73$«, yy, Invt. 


8-20 


9 


92 


R 


=5 


(71. 73, njx, yy 


8 (19.21) 


9 
10 


S93 
S93&W 


S,S' 


= 


(101, 102, 102, 101$ar, y)» 


9 (21.. 24) 


9 


94 


T 


S 


(190, 190$ar, y)' 


11 (27, 28) 


3 


29 


U 


= 


(252$ar, y)», Invt. 


12-30 


9 


95 


V 


= 


(325, 326$ar, yV 


13 (32, 33) 


5 


29a 


W 


= 


(967$a!, y)», Invt. 


18-45 



36—2 



284 



TABLES OF THE OOVARIANTS M TO W OF THE BINARY QUIMTIC. [Ud 



M. No. 83. 



cflfief 


• ■ • 


a» 6y 


• • • 


a* b^f* 


a' 6W» 


• • • 


o» 6y 


• • • 


b'eef* 


«•/ 


• • • 


Ved^* 


- 1 


dy - 1 


fiV/* 


- 1 


c^f 


+ 1 


A»/ + 2 


edef 


+ 6 


dV" 


+ 1 


«* - 1 


of 


- 3 


d^ 


- 1 


a' ft««/» 


dff 


- 3 


a» m/* 


+ 1 


b'ed/* + 6 


«?"«• 


+ 2 


«•/ 


- 1 


<!«■/- 5 


a' 6»c/» 


+ 2 


6'c«/« 


+ 1 


««V - 6 


A/ 


- 5 


cdef 


+ 6 


<fc» + 6 


«» 


+ 3 


of 


- 8 


bvy - 3 


6V?/" 


- 5 


dPf 


-10 


c»«fc/ + 7 


ccPf 


+ 7 


d^if 


+ 11 


«•«» + 2 


ed<? 


- 1 


bVef 


-10 


aP/ - 1 


dP« 


- 1 


<?dff 


+ 11 


«?«■ - 8 


h*(?dff 


- 1 


<?d^ 


+ 18 


<?« + 3 


<?^ 


+ 6 


ecPe 


-28 


a^6«i^ - 3 


<?dPe 


- 8 


(P 


+ 9 


«!/• + 3 


evf 


+ 3 


a<> 6»c/» 


- 1 


6V/» + 2 


a* b*/* 


- 1 


def 


- 8 


afo/ - 1 


Vcrf 


+ 5 


«> 


+ 9 


«»» - 3 


<Pf 


+ 2 


l^«/ 


+ 11 


dy + 6 


d^ 


- 3 


etPf 


+ 18 


«?«■ - 4 


Wdf 


- 8 


cdt? 


-37 


5'c««/ - 1 


«»«■ 


- 4 


<?« 


+ 8 


c'd'/ - 8 


cePs 


+ 7 


b'<*d/ 


-28 


cVfc* + 7 


# 


- 1 


c»«» 


+ 8 


wPe + 5 


6"cy 


+ 3 


c'd'e 


+ 37 


<i» - 3 


cida 


+ 6 


cd^ 


-17 


b'e'd/ + 3 


i?df 


- 4 


6V/ 


+ 9 


C«(J» - 1 


6V« 


- 3 


c^ 


-17 


c»<?« - 4 


c^d* 


- 2 


e'd' 


+ 8 


<!«# + 2 



5*, y)^ 



:fc 7 
SI 
84 



2 

67 

106 



^ 2 
22 
28 



^ 62 



db 167 



db 62 



143] 



TABLES OF THE 00VARIANT8 M TO W OF THE BINARY QUINTIC. 



285 









N. No. 84. 




a»6«(^ - 


1 


a» 6»«/» 


a« 6y» 


a« 6y» 


a« 6 V* + 1 


^/ + 


1 


a> h^df^ ~ 4 


a« 6»(5/» 


6»c«/« + 4 


deP - ^ 


1 a» h'cf^ + 


3 


«y + 4 


6»c(^ 


rfy* - 4 


«y + 2 


def + 


2 


6V/» + 4 


c«y + 6 


(i^y - 4 


a* 6y» - 1 


<5» - 


5 


edef - 8 


d««/ - 12 


«< + 4 


h'ceP - 2 


6V^ - 


8 


c«» + 4 


cie» + 6 


ai 6V' - 4 


d'P + 8 


cdy + 


2 


d^f 


a' 6"(^« - 6 


6*c(^« + 8 


cfey - 2 


cd^ + 


12 


cP«* 


ey 


c«y - 16 


«* - 6 


cP« - 


6 


a* 6"c/» + 4 


6»(y« + 12 


d'ef + 48 


6 V(^ - 2 


a^fty* - 


2 


def + 16 


cdef 


dfe» - 32 


c»«y + 6 


fe'c^/ - 


2 


«> - 24 


ce» - 36 


6V/« 


cd^ef- 20 


^/ - 


6 


h't^ef - 48 


cPf + 48 


c«cfe/- 40 


cdfe» + 12 


d^ + 


13 


ccP/ + 40 


d«c« - 12 


cV + 56 


rfy + 9 


6ic«e(/" + 


20 


ccfe» + 40 


6Ve/ - 48 


cd^f + 8 


rf»«« - 6 


c«c« + 


4 


d»« - 24 


c*gP/ ... 


c(?e» - 40 


a» 6»(5/* + 5 


CflPc - 


52 


h^(*df - 8 


<?d^ + 156 


rf*e + 12 


6«c(^« - 12 


rf* + 


24 


c»c« + 56 


ccPc - 168 


a» 6»(^« - 4 


cey - 13 


6V/ - 


9 


c>(?tf - 88 


rf» + 54 


ey + 24 


(/»«/ - 4 


C»(fe + 


20 


c^ + 36 


a« 6»c/» - 6 


6"cy» 


d8» + 15 


c*rf» - 


10 


ao 6y« - 4 


def + 36 


cflfe/" - 40 


6V/« + 6 


«*• 6 V + 


6 


h'cef + 32 


o • • . 


ce» - 60 


c«cfe/+ 52 


l^cdf + 


12 


d«/ - 56 


6V«/ + 12 


dy - 56 


c«c» - 10 


C6« - 


15 


cfe> + 60 


cd^f - 156 


d»e» + 100 


cdy - 20 


rf*c + 


10 


b'i^df + 40 


ccfe* 


6Vc/ + 24 


c(i»«« - 30 


6V/ + 


6 


c>e» -100 


d^e + 60 


c«dy+ 88 


d^e + 15 


c*cfc + 


30 


c<?e - 80 


6»c»e(/" + 168 


c«rfe» + 80 


6Ve/ - 24 


cd» - 


20 


rf* + 60 


c'c' - 60 


ciPe - 200 


c»rfy+ 10 


i^c^c - 


15 


6V/ - 12 


c'cPe 


d" + 60 


c»cfe> + 20 


c»rf» + 


10 


(*de + 200 


c(^ - 30 


6Ve(r - 36 


<?d^e - 10 


6Vrf 


• • • 


c«rf» -120 


6V/ - 54 


c*6» - 60 


ccP 






6Ve - 60 


c*(ig + 30 


c^d^e + 120 








c*(? + 40 


c'cP 


cW - 40 




di 


1 
19 


db 12 


:i= 12 


db 8 


db 8 




81 


192 


270 


182 


87 




62 


482 


806 


496 


128 



\^^ yY 



± 168 



db 686 



d. 588 



d= 686 



db 168 



286 



TABLES OF THE 00 VARIANTS M TO W OF TUB BINABT QUINTIC. [143 



0. 


No. 90. 


a» 6V/» + 1 


a» b^dp - 1 


dtp - 4 


«y* + 1 


«y + 3 


a« 6V' + * 


a« 6"/» - I 


(fo/* + 3 


b'ceP - 3 


^/ ~ 7 


d'P + 16 


b^t^eP - 16 


cfey + 4 


ccr/*+ 6 


«* - 15 


c<foy+ 30 


6 V(^« - 6 


ce* - 8 


c»cy + 4 


JV - 18 


c<i»c/- 22 


d»«' + 6 


cdg> + 26 


a* 6»/» - 3 


rf*/ + 9 


il'ceP - 4 


d»c« - 12 


(?/« - 4 


a» b^eP + 7 


cfey - I 


6«<^ - 30 


tf* + 18 


c^f + 1 


6^(4/^+ 22 


rf»c/ - 74 


c»e»/+ 74 


cfc» + 84 


c<^«/-160 


6'cy + 18 


«fo» - 32 


c'def + 160 


rf*/ + 81 


c'tf' - 98 


d»e» + 6 


e(P/ - 20 


fcV/* - 9 


cd'e' - 94 


(^de/^ 20 


c?'^ +51 


c»«» -112 


6Vc/ - 81 


c«dy- 18 


c»dy+ 18 


c«rf»«» + 284 


i^de" + 140 


cd'e -216 


C»d»<5 - 100 


d» + 54 


crf» + 18 


a*6*c/« + 15 


a» b'dP + 8 


b^cdp - 26 


e»/ - 18 


ce'/ - 84 


b'c'P - 6 


(?»«/ + 98 


crife/" + 32 


d^ 45 


c<5» + 45 


6V/« + 12 


d»/ + 112 


c'de/-^ 94 


(i»e» - 150 


c««' + 150 


6V«/ - 6 


cd*/ - 140 


c>rfy - 284 


ccP«« - 50 


c»cfe> + 50 


d'e + 15 


ccPe + 320 


b'c'ef - 51 


rf» -120 


c>d«/ + 100 


6»c*ej^ +216 


c»cfe« - 320 


cV - 15 


cW« + 310 


c«(i»« - 310 


crf» - 90 


e'd' + 130 


6Vrf/ - 18 


ftocy - 54 


c»«« + 120 


c^de + 90 


c*(?« - 130 


c*d» - 40 


c»rf* + 40 


:&: 4 


± 1 


69 


49 


497 


669 


1003 


954 



H^. yY 



1563 



dbl563 



8] TABLBS OF THE COVARIANTS M TO W OF THE BINABT QUINTIC. 



287 











P. No. 91. 








• • • 


a» Vf* 


a»6V' - 


1 


a« h^df^ + 1 


a» b^eP 


• • • 


c^ b^P 


• • • 


VceP - 2 


deP + 


6 


^P - 1 


a* bHp + 


2 


a^b^ep 


+ 1 


dy« + 6 


^/ - 


5 


a« b'cP - 6 


cy« - 


2 


b^cdp - 1 


- 2 


d^f - 1 


a« 6y« + 


1 


d^ + n 


b^^p - 


5 


c«y+ 1 


+ 2 


tf* - 2 


6^C<5/« - 


11 


ey - 5 


edeP^ 


17 


(PeP+ 3 


- 1 


a* \?eP + 2 


d^P - 


4 


6Ve/« + 4 


c^f - 


7 


flfey - 5 


- 1 


h^cdp - 17 


(^/ - 


4 


cd'P- 2 


d^P -^ 


4 


e» + 2 


+ 2 


c^f + 13 


c* + 


17 


c(iey+ 4 


d^e^f- 


6 


a} b^df^ + 2 


- 3 


^tf - 32 


y^&dp^ 


2 


cc* — 4 


d^ + 


5 


ey« - 2 


- 6 


cfe» + 32 


c»«y + 


26 


d'ef - 10 


a* 6V* + 


1 


h'ifp - 2 


+ 13 


6V/« + 4 


c^ef- 


2 


rf'tf' + 8 


cfo/« - 


13 


cdtp^ 6 


- 8 


c»cfe/+ 36 


cd^ - 


40 


a' 6y» + 5 


^/ + 


12 


cey - 2 


+ 2 


c»«» - 24 


rfy - 


9 


b^cep + 4 


6'cV^ + 


32 


d^P - 16 


+ 16 


cd^f - 10 


rPe' + 


24 


(^/« - 26 


cd^P^ 


36 


d«cy + 24 


- 2 


oi»«» - 16 


a} h'ep + 


5 


de"/ - 35 


cd^f- 


42 


rfe* - 10 


-38 


d^e + 12 


6'crf/« - 


4 


«< + 42 


ce* + 


24 


6V<5/« + 8 


+ 34 


a> hHp + 7 


ct^f + 


35 


b'c^dp + 2 


cP^ + 


56 


(?d^P^ 2 


- 9 


ey - 12 


rf*?/" - 


26 


cV/+ 26 


cP«» - 


34 


c'cfey- 52 ! 


+ 5 


6V/« + 6 


efo« - 


22 


cgPc/" + 72 


6V^« + 


10 


cV + 28 


+ 2 


cdef + 42 


6'cy + 


10 


cde" - 124 


c»cy- 


54 


cd?ef ^ 52 j 


- 12 


C6^ • • • 


c'def- 


72 


rfy + 13 


c^d^ef+ 


64 


ccPe^ - 32 


-24 


e^f + 54 


cV - 


106 


d"^ + 26 


c^d^ + 


46 


d^f - 18 


+ 52 


(f«e« - 91 


ccP/ + 


76 


6^cy* + 9 


cd'f - 


37 


rfV + 12 1 




6Vc/ - 68 


ccPc* + 


210 


cW«/- 76 


cd^e^ - 


50 


a« 6V' + 1 ' 


-22 


c»rfy_ 64 


d^'e - 


99 


cV - 56 


cPe + 


21 


ei^/« - 13 i 


-52 


c«cfe« + 14 


¥c*ef - 


13 


c«c^/+ 10 


a« 6y» + 


2 


ey + 12 


+ 34 


ccPc + 204 


i?d^f- 


10 


c^cTe^ + 296 


6»c^/» - 


32 


b^c^eP - 2 


+ 8 


^ - 93 


i^de" + 


128 


cd'e -260 


c^y^ + 


24 


ccP/- + 38 


- 1 


V'i^df + 37 


c'c^e - 


184 


c^» + 72 


d^f 


• • ■ 


crf«y- 7 1 


+ 18 


cV + 86 


cc? + 


72 


a' b'ep - 17 


e* 


• • • 


ce* - 30 


-25 


c*(^c - 208 


a« 6*c(/^ + 


4 


6»c4/'« + 40 


6V(y« + 


16 


d^ef - 34 


+ 10 


c«rf^ + 86 


^/ - 


42 


cey + 22 


c^^y + 


91 


d^^ +35 


- 2 


a« 6^c/« - 5 


6V/« - 


8 


d"^ + 106 


cd^ef- 


14 


6Ve(/^ - 34 


+ 10 


cfe/ - 12 


cd«/* + 


124 


(ie^ -105 


cd^ - 


105 


c»ey + 22 


-28 


c • • • 


cc* + 


105 


6V/« - 24 


d^f - 


86 


c«rf«c/- 8 


+ 30 


6Ve/ + 34 


rfy + 


56 


c«(fe/-210 


cPc« + 


110 


i?df? + 50 


+ 32 


crfy - 46 


C^6» - 


130 


c»c» + 130 


6V/« - 


12 


ccfy + 25 


-35 


cd^ + 105 


6V«/ - 


26 


cd"/ - 128 


i^d^f- 


204 


cdJ'^ - 70 


-50 


c^e - 20 


<?d^f- 


296 


cd"^ + 170 


i?^ + 


20 


rf*c + 15 1 


+ 30 


l^i?df + 50 


i?d^ - 


170 


rf^e - 25 


(^d^f + 


208 


6V/« + 9 


- 12 


c»c« -110 


ccPe + 


340 


b'c*e/ + 99 


c«c^e« + 


170 


c*<fe/+ 1 


+ 70 


c*ci»e - 170 


c/» - 


60 


c»rfy + 184 


cc?*c - 


250 


cV -30 


-40 


cd?' + 115 


h'i^df + 


260 


c»c£e« - 340 


^ + 


60 


c'ciy - 10 


-15 


6V/ - 21 


cV + 


25 


c'd/'e + 150 


6Ve/ + 


93 


c'(^c« + 40 


+ 10 


d'dA + 250 


c'cPe - 


150 


ccf - 40 


c'd^f- 


86 


<?d^e - 10 


> • a 


c«rf» - 150 


c«c^ 


« • ■ 


b'c'df - 72 


^d^ - 


115 


C€^ 




6V« - 60 


6«cy - 


72 


c»c« + 60 


c^cPc + 


150 






&d^ + 40 


c"cfe + 


40 


c*d^e 


cW - 


40 






• 


c^d^ 


« • • 


c'd* 








± 3 


=k= 6 


Jt^ 


6 


^ 1 








67 


99 




70 


27 


± 


24 


± 6 


136 


536 




536 


577 




266 


134 


182 


594 




954 


961 


« 


944 


248 



5«, uf 



^ 388 



dbl234 



:tl566 



rtl566 



±1234 



db38a 



288 



TABLES OF THE CO VARIANTS M TO W OF THE BIKABT QUINTIC. [l43 



Q. No. 25. Q'. No. 26. 



Q. No. 25. Q'. No. 26. 



a« bof* 


• • • 


+ 1 


a» 6V 


+ 27 


- 3375 


a* b'ef* 


• • • 


20 


h^t^dp 


- 48 


-1- 5760 


b'cd/* 


+ 1 


- 120 


<?i?f 


+ 3 


- 600 


c<?P 


- 1 


+ 160 


cd^ef 


+ 106 


- 16000 


(Pep 


- 3 


+ 360 


ed^ 


- 81 


+ 9000 


d^f 


+ 5 


- 640 


d^f 


- 38 


+ 6400 


«• 


- 2 


+ 256 


rf»c« 


+ 38 


- 4000 


a* b'df* 


- 1 


+ 160 


6V/« 


+ 18 


- 2160 


«•/* 


+ 1 


10 


i^def 


- 30 


+ 7200 


W/» 


- 3 


+ 360 


&^ 


+ 38 


- 4000 


cdeP 


+ 11 


- 1640 


t^d^f 


+ 8 


- 3200 


<^f 


- 5 


+ 320 


<?d^^ 


+ 25 


+ 2000 


d»/« 


+ 12 


- 1440 


cd^e 


- 57 


• • • 


c?^f 


- 30 


+ 4080 


rf» 


+ 18 






<fe« 


+ 15 


- 1920 


b'i^ef 


- 9 






6V«/» 


+ 12 


- 1440 


c*d*/ 


+ 6 






<?cPp 


- 21 


+ 2640 


C*(fo« 


- 57 






<?d<?f 


- 34 


+ 4480 


c'd'e 


+ 74 






<?^ 


+ 22 


- 2560 


c'd* 


- 24 






c^rf 


+ 78 


- 10080 


y'cHf 


• • • 






edP^ 


- 48 


+ 5760 


c«c« 


+ 18 






^f 


- 27 


+ 3456 


c^cf'c 


- 24 






<?«» 


+ 18 


- 2160 


c*rf* 


+ 8 






a' VcP 


+ 5 


- 640 








d<P 


- 5 


+ 320 








«y 


. • • 


- 180 








l^<?ep 


- 30 


+ 4080 








eePP 


- 34 


+ 4480 








edff 


+ 133 


- 14920 








ct^ 


- 54 


+ 7200 








«P«/ 


- 18 


+ 960 








dP^ 


+ 3 


~ 600 








b^i*dp 


+ 78 


- 10080 








e^f 


- 18 


+ 960 








<?d^ef 


- 220 


+ 28480 








<»d<? 


+ 106 


- 16000 








cd>f 


+ 93 


- 11520 








c<i*«s* 


- 30 


+ 7200 




Thesm 


018 for Q' are 


d*e 


- 9 


• * . 




1 


= 1 


b'>(*P 
c*def 


- 27 
+ 93 


+ 3456 
- 11520 




776- 
21266- 
68656- 


- 780= -4 

- 21250= +6 

- 68660= -4 


«!*«• 


- 38 


+ 6400 




87816- 


- 87815= +1 


<?d^f 
c'rf's* 


— 4.9 


+ 5120 
- 3200 








+ 8 




128505- 


-128505= 


c'd'e 


+ 6 


• • • 








cd" 


. • • 


• • • 








a* 6»/» 


- 2 


+ 256 








6V^ 


+ 15 


+ 1920 








«P/» 


+ 22 


- 2560 








d^f 


- 54 


+ 7200 
















± 6 


til 128505 










169 












525 












424 







^1124 



3] TABLES OF THE COVARIANTS M TO W OF THE BINARY QUINTIC. 



289 









R. 


No. 92. 










• • • 


a» Vc^f - 15 


a* 6y* 


• • • 


a^ 6V 


• • • 


o» jy 


• » • 


a« 6»d»e/ + 2 


• a • 


rf«e/- 38 


a' 6 V* 


• • • 


6Vc^ + 


18 


Veep 


• • . 


dJ'd' + 15 


• • • 


cfe* + 46 


h'cdp 


• • • 


i^^f - 


66 


dy« + 


1 


6Vrf/«- 32 


- 1 


6»cy« + 3 


c^P 


• • • 


cd^tf ■\- 


20 


d4?P- 


2 


c»e»/ - 39 


+ 6 


c«(fe/+ 102 


(^e/« + 


2 


cd^ 


• • • 


«y + 


1 


cWe/- 24 


- 4 


c»«» - 16 


cfey - 


4 


d*f + 


58 


a* 6V' 


• • • 


c«dfe» + 175 


- 3 


cdy+ 76 


6» + 


2 


d'e' - 


50 


l^cdp - 


6 


c^/ + 25 


+ 1 


crfV - 175 


a» }^dp 


• • • 


6V/' - 


6 


c^P + 


6 


cd»c« - 120 


+ 1 


d^e + 35 


^P 


• • • 


c^def -h 


72 


cPeP + 


3 


c^e + 15 


+ 2 


6*cV - 42 


h^i?P - 


2 


cV + 


50 


d4?f 


• • • 


6V/« + 9 


- 6 


c»(^/~ 182 


cdeP 


• • • 


c^d^f- 


156 


«• - 


3 


d'def + 106 


+ 4 


i?d^ + 120 


ct^f + 


4 


c«rf*e» 


• • • 


6V/' + 


3 


c*e» - 35 


- 3 


c>d»« + 150 


d»/» - 


14 


cd^e + 


90 


<?deP- 


3 


c»dy- 60 


+ 3 


c(i» - 70 


d^^f + 


30 


d» - 


30 


<?«*/- 


6 


c»d«e»-150 


- 18 


h^&df + 126 


d^ - 


18 


6V^/ - 


24 


cePp 


• • • 


c»rf*c + 176 


+ 17 


&^ - 15 


6Vc/« + 


14 


cW + 


94 


cd^^f-h 


3 


crf« - 45 


+ 22 


c*d»« - 175 


cW/« 


• • • 


i^d^ - 


90 


c<fo* + 


6 


6Vc/ - 36 


- 21 


(?d* 4- 75 


(^d^f- 


66 


c»^e 


• • • 


#«/ 


• • • 


d'd^f^ 21 


• • • 


6V/ - 27 


(?^ + 


26 


c«c?» + 


10 


<?«• - 


3 


c»rf6> + 70 


+ 13 


c«(fe + 45 


cc^ef + 


56 


h^d'df - 


18 


a' 6'd/» + 


4 


c*d»« - 75 


- 12 


c»(^ - 20 


cd^^ - 


18 


C«6» + 


30 


«•/' - 


4 


c>d" + 20 


- 21 




^f - 


18 


<^d^e ~ 


10 


6V/» - 


1 




- 3 




d^^ + 


6 


c^c^^ 


• • • 


c<fo/^ + 


18 




+ 32 




a» 6»c/« + 


4 






w*/ - 


16 




- 9 




dep - 


4 






ePP - 


13 




- 1 




^/ 


• • • 






cP^/- 


3 




• • • 




6 V«/« - 


30 






d^ + 


15 




+ 6 




cd^P^- 


66 






h^<?fp - 


22 




+ 16 




cde^f 


• • • 






c'cPP* 


12 




- 18 




c^ - 


18 






(?d4?f+ 


18 




- 3 




d^ef - 


84 






c*e* + 


38 




+ 3 




d^iS" + 


66 






C€Pef + 


32 




- 18 




h'i^dP-- 


56 






cdP^ - 


102 




+ U 




c^^f ^ 


84 






<Pf - 


18 




- 41 




^d^tf 


% » * 






d**" + 


42 




+ 39 




c^d^ - 


20 






Ve^dp + 


3 




• • • 




cd^f + 


40 






cV/ + 


41 




- 32 




cd^^ - 


72 






<?cPef- 


84 




- 2 




(Pe + 


24 






<?d^ - 


76 




+ 84 




6V/« + 


18 






c'd'/ + 


33 




+ 24 




c*de/- 


40 






«•<?«• + 


182 




-106 




c^e» - 


58 






cd»« - 


126 




+ 36 




c'dy 


• • • 






rf' + 


27 




+ 18 




c'cPe' + 


156 






a'iV* - 


1 




~ 33 


± 8 


c«c^e - 


94 


± 4 




deP - 


17 


:&: 2 


- 25 


98 


cc?* + 


18 


136 




«y + 


18 


21 


+ 60 


300 


a' b'P - 


2 


476 




6Ve/» + 


21 


465 


- 21 


780 


6*c«/^ + 


18 


478 




cd>/« + 


21 


693 


+ 3 


±1181 


d'P - 


26 


:tl094 




cd^f- 


14 


:lrll81 


- 6 




d^f + 


18 






ce* — 


45 





\«^ y)'- 



C. II. 



37 



290 



TABLES OF THE C0VABIANT8 M TO W OF THE BINABY QUINTIC. [l43 



S. No. 93 6m ; S'. No. 93. (•5«. y)». 



CJoof. *• 


S 




• 


3' 


Coef.x» 


S 


8' 


Coef. j:*// 


s 




8' 


Coef. £*y 


8 


V 


a* by* 








• • • 


a} l^d'ef 


- 66 


+ 528 


a* 6V* 




+ 


9 


a> b't?^ 


+ 6& 


Vcef* 






+ 


9 


d^e" 


+ 72 


45 


deP 




— 


45 


C(Pef 


+ 7B\ 


cpr 






+ 


21 


h^(?dp 


- 21 


- 2592 


^P 




+ 


36 


c^^ 


- 1&& 


d^r 






— 


78 


C»6>/ 


- 96 


- 9747 


a» 6y* 




— 


9 


d^f 




c*/ 






+ 


48 


ed^ef 


+ 36 


- 8496 


6W* 




— 


18 


d*«« 


V 


cfVeP 






— 


9 


^d^ 


+ 213 


+ 26610 


d^P 




+ 


243 


b'd'dp 


+ L ^*- 


h^cdp 






— 


162 


cd^f 


+ 120 


+ 8544 


d^P 




+ 


9 


cV/ 


__ ^^ 


c<?P 






+ 


99 


C(P^ 


- 303 


- 16650 


eV 




— 


216 


<?d^tf 


— 


dPtP 






+ 


309 


d^e 


+ 51 


+ 720 


V'&dp 


- 3 


— 


351 


C»d8» 


— 


A*/ 






+ 


12 


h'^P 


+ 9 


+ 972 


i^^P 


+ 3 


+ 


144 


c*d^f 


— 


«• 






— 


240 


^def 


+ 174 


+ 24624 


cdPeP 


+ 24 


+ 


1836 


i^d^i? 


+ 1 - 


6V/» 


— 


2 


— 


81 


cV 


- 36 


- 5040 


cd^f 


- 42 


— 


2592 


cd^e 


+ 


<?der 


+ 


15 


+ 


1026 


i^d^f 


- 204 


- 15984 


C6» 


+ 18 


+ 


1152 


d' 


— 


<!»«•/ 


— 


9 


— 


768 


c»e?«» 


- 174 


- 29340 


d^P 


- 18 


— 


1458 


W/« 


— 


eeP/* 


— 


9 


— 


738 


(^d^e 


+ 330 


+ 34320 


(Pt?f 


+ 33 


+ 


2268 


<^def 


+ 


«?«»/ 


— 


6 


— 


664 


cd^ 


- 99 


- 8640 


(i»«* 


- 15 


— 


1008 


c««» 


+ 


ede* 


+ 


9 


+ 


1056 


6V(5/ 


- 63 


- 7776 


a^h*ip 


• • • 


+ 


63 


<M^f 


+ 


d*ef 


+ 


9 


+ 


756 


^d^f 


+ 66 


+ 6184 


b'cdp 


+ 6 


— 


234 


cV?«« 


• 


<?«» 


— 


7 


— 


696 


<^d4^ 


+ 99 


+ 12960 


cep 


- 6 


— 


18 


<?d^ 


+ 


o?Vdp 




• • • 


+ 


120 


d^e 


- 147 


- 14400 


d*ep 


- 24 


— 


3231 


c'd' 


+ 


«•/• 




• • • 


— 


21 


c>d» 


+ 45 


+ 3840 


^f 


+ 42 


+ 


4293 


a* ft»c/» 


+ _ 


v<»p 


+ 


6 


+ 


486 


a* 6y • 


+ 2 


+ 192 


^ 


- 18 


— 


972 


deP 


— ^^ 


edef* 


— 


30 


— 


2160 


h^ceP 


- 15 


- 1440 


h^i?P 


+ 3 


+ 


810 


^f 


+ tfS 


c^f 


+ 


18 


+ 


1023 


d^r 


- 6 


^ 192 


<^dep 


- 78 


— 


3826 


h^eP 


.^ ^ 


d}/* 


+ 


9 


+ 


120 


*y 


- 18 


- 1080 


<?i?f 


+ 69 


+ 


4032 


cdy* 


+ Sf 


d^f 


+ 


6 


— 


1053 


<5* 


+ 27 


+ 2025 


cd^P 


+ 93 


+ 


7938 


cd^f 


- 9^ 


(fe« 


— 


9 


+ 


1314 


b'c'tif* 


+ 24 


+ 1728 


cd^^f 


- 61 


— . 


9360 


c«* 


- zr' 


h^eP 


— 


15 


— 


1863 


i?^f 


+ 61 


+ 4410 


ede' 


- 33 


— 


864 


d^ef 


^r 60* 


<?€Pr 


+ 


21 


+ 


2538 


cd^ef 


+ 102 


+ 5280 


d>ef 


- 57 


— 


1296 


(?«» 


- 45 


<?def 


— 


6 


+ 


2340 


cd^ 


- 171 


- 13500 


(Pe" 


+ 54 


+ 


2700 


h^<^dp 


- 39V 


<^e* 


+ 


18 


+ 


672 


^f 


+ 6 


- 4800 


h\^eP 


+ 24 


— 


324 


t^^f 


+ 45 


'^ 


ed^ef 


+ 


30 


+ 


2820 


d^i? 


+ 18 


+ 7800 


i?d^P 


- 36 


— 


2484 


i^d^tf 


- 108 




«?«• 


— 


61 


— 


7812 


l^c'P 


- 9 


- 648 


i^d^f 


- 9 


+ 


6624 


c>cfe» 


+ 96 


r- 


dy 


— 


36 


— 


3024 


&def 


- 210 


- 14040 


c»«* 


- 64 


— 


6912 


cdyf 


- Ill 


^ 


d*** 


+ 


39 


+ 


4672 


<*^ 


+ 43 


+ 3076 


i^d^tf 


+ 24 


— 


4428 


edP^ 


+ 147 


— 


JVt^' 


— 


3 


— 


324 


<?d^f 


- 120 


+ 9120 


c>d»«> 


+ 129 


+ 


12672 


d^t 


- 30 


+ 


c'e'/ 


+ 


46 


+ 


3888 


C>(?«' 


+ 345 


+ 16350 


cd^f 


+ 9 


+ 


1944 


6V/« 


+ 9 


+ 


i^d'ef 


— 


84 


— 


8748 


cd^t 


- 87 


- 19200 


cc?*e' 


- 114 


— 


9072 


cW 


+ 6 


— 


c'de* 


— 


63 


— 


4800 


d" 


- 2 


+ 4800 


d^e 


+ 27 


+ 


1944 


<?*«> 


- 48 


— 


^d^f 


+ 


45 


+ 


4248 


IMef 


+ 72 


+ 4860 


a^h^dp 


- 3 


+ 


144 


i^d^f 


+ 234 


- 


<?dPt? 


+ 


150 


+ 


14620 


d'd^f 


+ 240 


- 3240 


^P 


+ 3 


— 


243 


i?d^^ 


- 150 


-» 


(xPe 


— 


117 


— 


11448 


(H^ 


- 192 


- 8100 


b'i^P 


- 6 


— 


900 


C\^J 


- 108 


- 


cP 


+ 


27 


+ 


2692 


(*d^e 


- 186 


+ 9000 


edeP 


+ 108 


+ 


10620 


erf* 


+ 57 




a> b*e/* 


— 


6 


— 


576 


c»d» 


+ 96 


- 2400 


c/f 


- 96 


— 


8586 


h^^tf 


+ 9 




def* 


+ 


15 


+ 


672 


h^d'df 


- 144 




d^P 


~ 21 


— 


864 


i?d^f 


- 141 




«•/ 


— 


9 


— 


459 


C«6> 


+ 18 




d^^f 


- 48 


— 


1215 


&d^ 


+ 87 




V<?eP 


+ 


30 


+ 


3466 


i^(Pe 


+ 201 




df^ 


+ 63 


+ 


1215 


e^e 


+ 96 




cd^r 


— 


15 


— 


864 


c'd^ 


- 87 




6V^ 


- 24 


— 


1836 


i^d" 


- 51 




ed^f 


+ 


24 


+ 


2094 


h'<?f 


+ 27 




(?d^P 


- 123 


— 


16812 


b^e'd/ 


+ 27 




C9* 




45 




3915 


i?dt 
i^d^ 


- 46 
+ 20 




cW/ 


+ 147 


+ 


6651 




- 18 

- 21 
+ 12 





For the Namerical Verifications for S see 
further pp. 304, 806. 



78 

db as 3258 

414 41253 

1284 124524 

1292 68640 

db8028 :k 287758 



± 78 
480 
927 
96 6 

db245i 



143] TABLES OF THE OOVARIANTS M TO W OF THE BINARY QITINTia 



291 



S. No. 93 Us; S'. No. 93. 



f« 


s 




8' 


Coef. xy' 




S 




8' 




Ck>ef. y* 


S 




S' 


Coef. y» 


S 


8' 




• • • 


_ 


9 


a^b'i?^ 


_^ 


60 


+ 


4320 


a* 6 V* 






• • a 


a^ b^d'd 


- 72 


- 4860 


1 


• • • 


+ 


9 


i^ef 


+ 


36 


+ 


14544 


a» b'd/' 




— 


9 


b^'deP 


+ 36 


+ 3024 




• • 


+ 


45 


i^dPi^ 


+ 


108 


— 


3060 


^P 




+ 


9 


dd^P 


~ 45 


- 4248 


t 


• • • 


+ 


18 


c^f 


— 


24 


— 


5184 


¥df* 




— 


21 


dddf 


- 120 


- 8544 


1 


• • • 


— 


63 


cd"^ 


— 


6 


+ 


1620 


cdep 




+ 


162 


dd 


- 6 


+ 4800 


•B 


• • • 


— 


243 


d'e 


— 


9 




• • • 


cdp 




— 


120 


dd^ef 


+ 204 


+ 15984 


r» 


+ 3 


+ 


351 


h^<*dp 


— 


9 


— 


1944 


rfy» 


+ 2 


+ 


81 


d(Pd 


+ 120 


- 9120 


/• 


- 6 


+ 


234 


i^i^f 


— 


51 


+ 


3888 


d}dp 


- 6 


— 


486 


dd^f 


- 66 


- 5184 


■ 


+ 3 


— 


144 


d'd^ef 


+ 


96 


— 


1296 


de*f 


+ 6 


+ 


576 


dd'd 


- 240 


+ 3240* 


1 


- 3 


— 


810 


C*d8» 


+ 


111 


— 


1440 


d 


- 2 


— 


192 


cd^e 


+ 144 


• • • 


f 


+ 6 


+ 


900 


f?df 


— 


27 


+ 


576 


a« h'cP 


• • • 


+ 


78 


<f» 


- 27 


• • • 




- 3 


■— 


288 


c»d»«« 


— 


234 


+ 


360 


dtp 


• • • 


— 


99 


d' b^ep 


• • • 


+ 240 




• • • 


— 


36 


c«d»« 


+ 


141 




• • • 


^P 


• • • 


+ 


21 


b^cdP 


- 9 


- 1056 


1 


• • • 


— 


9 


erf' 


— 


27 




• • • 


h^deP 


• • • 


— 


309 


cdp 


+ 9 


- 1314 


1 


- 3 


— 


144 


a^l^dp 


— 


18 


— 


1152 


cd^P 


- 15 


— 


1026 


d^eP 


- 18 


- 672 


ri 


+ 6 


+ 


18 


^P 


+ 


18 


+ 


972 


cddp 


+ 30 


+ 


2160 


ddf 


+ 45 


+ 3915 




- 3 


+ 


243 


6V/« 


+ 


15 


+ 


1008 


cdf 


- 15 


— 


672 


d 


- 27 


- 2025 


1 


- 24 




1836 


cdep 


+ 


33 


+ 


864 


d'ep 


+ 15 


+ 


1863 


l^dp 


+ 7 


+ 696 


n 


+ 24 


+ 


3231 


c^f 


— 


63 


— 


1215 


^df 


- 30 


— 


3456 


ddeP 


+ 51 


+ 7812 


r 


+ 78 


+ 


3825 


(PP 


+ 


54 


+ 


6912 


dd 


+ 15 


+ 


1440 


ddf 


- 72 


+ 45 


f 


- 108 


— 


10620 


d^^f 


— 


66 


— 


12960 


V'ddp 


+ 9 


+ 


738 


cd^P 


+ 63 


+ 4800 




+ 30 


+ 


3888 


de' 


+ 


27 


+ 


6075 


ddp 


- 9 


— 


120 


cd^df 


- 213 


- 26610 


1 


- 24 


+ 


324 


6V«/» 


— 


54 


— 


2700 


dd'eP 


- 21 


— 


2538 


cdd 


+ 171 


+ 13500 


f 


+ 24 


+ 


1836 


i»d^P 


— 


129 


— 


12672 


dddf 


+ 15 


+ 


864 


d^ef 


+ 36 


+ 5040 




• • • 


— 


756 


C'd4?f 


+ 


186 


+ 


18900 


dd 


+ 6 


+ 


192 


(Pd 


~ 43 


- 3075 


1 


+ 18 


+ 


1458 


c>«* 


+ 


45 


— 


6075 


cd^p 


+ 3 


-4- 


324 


l^deP 


- 39 


- 4572 


/' 


- 93 


— 


7938 


cd^ef 


+ 


54 


+ 


12960 


cd'df 


+ 21 


+ 


2592 


dd'p 


- 150 


- 14520 


f 


+ 21 


+ 


864 


cd^(^ 


— 


96 


— 


10125 


cd^d 


- 24 


— 


1728 


dddf 


+ 303 


+ 16650 


r 


+ 36 


+ 


2484 


d^f 


~ 


54 


— 


5760 


d^ef 


- 9 


-^ 


972 


dd 


- 18 


- 7800 


«y 


+ 123 


+ 


16812 


rf*e» 


+ 


48 


+ 


4500 


d^d 


+ 9 


+ 


648 


dd^ef 


+ 174 


+ 29340 


4 


- 51 


— 


7488 


l^^P 


+ 


114 


+ 


9072 


a' b*P 


• • • 


— 


48 


dd*d 


- 345 


- 16350 


f 


- Ill 


— 


15228 


c*«y 


+ 


9 


— 


2970 


b^ceP 


• • • 


— 


12 


c^f 


- 99 


- 12960 


t 


+ 39 


+ 


7128 


f?d^ef 


— 


150 




22500 


d'P 


+ 9 


+ 


768 


cd'd 


+ 192 


+ 8100 




+ 27 


+ 


.3888 


dd^ 


— 


147 


+ 


13950 


ddP 


- 18 


— 


1023 


d^e 


- 18 


• • • 




- 9 


— 


1944 


dd^f 


+ 


93 


+ 


9360 


ey 


+ 9 


+ 


459 


b'ddp 


+ 117 


+ 11448 




• • • 


+ 


216 


d^d 


+ 


150 


— 


6300 


l^ddp 


+ 6 


+ 


564 


ddf 


- 51 


- 720 


1 


+ 42 


+ 


2592 


ed^e 


— 


87 




• • • 


ddp 


- 6 


+ 


1053 


dd'ef 


- 330 


- 34320 


•a 


- 42 


— 


4293 


d? 


+ 


18 




• • • 


cd^eP 


+ 6 


— 


2340 


ddd 


+ 87 


+ 19200 


1 


- 69 


— 


4032 


b'dp 


— 


27 


— 


1944 


cddf 


- 24 


— 


2094 


dd^f 


+ 147 


+ 14400 


• 


+ 96 


+ 


8586 


ddef 


— 


30 


+ 


6480 


cd 


+ 18 


+ 


1080 


dd/'d 


+ 186 


- 9000 




- 27 


— 


3645 


dd 


+ 


30 


— 


3600 


d>p 


- 45 


— 


3888 


dd^e 


- 201 


• • • 




- 33 


— 


2268 


dd^f 


— 


6 


— 


2880 


d^df 


+ 96 


+ 


9747 


ccT 


+ 45 


• • • 


/^ 


+ 51 


+ 


9360 


d^d 


+ 


108 


+ 


1800 


d'd 


- 51 


— 


4410 


b'dp 


- 27 


- 2592 


r 


+ 48 


+ 


1215 


dd^e 


— 


96 




• • • 


b'dp 


- 9 


— 


756 


ddef 


+ 99 


+ 8640 


rt 


+ 9 


— 


6624 


dc? 


+ 


21 








ddep 


- 30 


— 


2820 


dd 


+ 2 


- 4800 


y 


- 147 


— 


6651 


b^def 


+ 


27 








• ddf 


+ 66 


— 


528 


dd^f 


- 45 


- 3840 




+ 39 


+ 


4050 


dcPf 


— 


9 








ddJ'P 


+ 84 


+ 


8748 


dd^d 


- 96 


+ 2400 


• 


+ 78 


+ 


4968 


ddd 


— 


57 








dd^df 


- 36 


+ 


8496 


dd^e 


+ 87 


• • • 




- 45 


— 


2970 


dd^e 


+ 


51 








ddd 


- 102 


— 


5280 


dd^ 


- 20 


• • • 


•1 


+ 57 


+ 


1296 


d(P 


— 


12 








cd^ef 


- 174 


— 


24624 








/• 


- 24 


+ 


4428 












• cd^d 


+ 210 


+ 


14040 








y 


- 78 


— 


18612 












d^f + 63 


+ 


7776 











He 9 


=k 12 


1548 


426 


45999 


912 


62019 


1101 


92853 



=b 8 


db 828 


123 


10920 


1071 


79779 


1821 


146226 



:fe2451 ±202428 



±8023 ±237753 



37—2 



292 



TABLES OF THB OOVABIAMTS M TO W OF THE BINAirr QUINTIC. [14 



( 











T. No. 94. 












X coe£Scient 










a coefficient 




a* VcP 




a'lt'^P 


_ 


20 


a> ft«c»d'«/» 


+ 


153 


a« 6V/« - 


6 


def* 




d'^P 


+ 


33 


<?d^f 


— 


390 


c»cfe/« + 


240 


«•/' 




<foy 


— 


48 


cV 


— 


234 


«•</■ + 


179 


a* bV 




«• 


+ 


27 


ed*P 


— 


114 


c»d»/« - 


144 


b'cef* 




V&fp 


+ 


39 


cd*<?f 


— 


308 


c'cPe'/ + 


306 


<P/* 




<?d}p 


— 


105 


c«P«* 


+ 


735 


c'd^ - 


765 


de>P 




<?d^P 


+ 


18 


d^ef 


+ 


208 


cct^f + 


28 


e*/* 




<?t^f 


— 


6 


d*e» 


—- 


283 


cd"«» + 


280 


VtMf* 


- "i 


ed?eP 


+ 


114 


wp 


+ 


27 


<p/ - 


88 


<?^P 


+ 1 


edPtff 


~ 


67 


<*deP 


— 


396 


(PeF + 


40 


ccPef* 


+ 7 


edtf 


+ 


12 


c'e'/ 


— 


337 


6"c»«/« - 


63 


cd4?p 


- 12 


dPp 


— 


6 


<?d^P 


+ 


222 


c\iPP + 


42 


c^f 


+ 5 


d'e'/ 


+ 


3 


<^<P^/ 


+ 


783 


<?*cfe>/ - 


798 


#/> 


- 6 


d*»^ 


— 


12 


c«<fo« 


+ 


880 


cV + 


176 


cP^P 


+ 12 


l^d'dP 


+ 


90 


c'd^ef 


+ 


93 


d^iPef - 


224 


dfi^f 


- 7 


<!««•/» 


— 


198 


e'd'e* 


— 


1986 


c»rf»«» + 


1366 


d^ 


+ 1 


(?d}eP 


— 


9 


cdff 


— 


240 


cW/ + 


368 


c? i'e/" 


• • • 


<?d^f 


+ 


238 


cd>^ 


+ 


1098 


c»d*«« - 


1026 


Vcdp 


+ 2 


<!»«• 


+ 


116 


d'e 


—- 


144 


cd^e + 


60 


c^P 


- 2 


^dfp 


— 


6 


b'ifcP 


+ 


81 


d» + 


30 


6?tP 


- 7 


i?d*^f 


+ 


108 


<fd'P 


— 


54 


l^^dp 


• « • 


d<?P 


+ 12 


c**?** 


— 


613 


(fde'f 


+ 


570 


if^f + 


252 


«•/ 


- 6 


cdPef 


— 


294 


c»«« 


— 


148 


c»c?«/ + 


798 


6'cy« 


+ 3 


edV 


+ 


513 


e*(Pe/ 


— 


1116 


C»d8» - 


700 


<?deP 


- 30 


d'/ 


+ 


108 


c*d'e* 


— 


627 


cV/ - 


578 


i?^P 


+ 21 


(?«• 


— 


153 


c«d»/ 


+ 


474 


iN^i? - 


370 


ecPP 


+ 44 


bfd'p 


— 


27 


<?d>^ 


+ 


1662 


c'cPtf + 


880 


edP«?P 


- 69 


<?deP 


+ 


108 


<?d^e 


— 


1185 


<NP - 


240 


«fey 


+ 62 


c»«'/ 


+ 


194 


eeP 


+ 


243 


6V/« 


... 


c«« 


- 28 


«*</•/* 


— 


42 


iVdp 




• • • 


<?def - 


486 


d^eP 


- 6 


c**?**/ 


— 


663 


c'ey 


— 


216 


<?^ + 


60 


d»e'/ 


- 8 


<*dif^ 


— 


274 


(fd'f/ 


+ 


369 


«•(/»/ + 


312 


<?«• 


+ 11 


ed^tf 


+ 


570 


d'd^ 


+ 


340 


c«dV + 


645 


6Ve/» 


- 6 


€»<?«• 


+ 


914 


(I'd*/ 


— 


149 


(*d^e - 


735 


ifd^P 


- 11 


<?dff 


— 


163 


(!»(?«" 


— 


730 


e^ + 


190 


<?d^P 


+ 96 


cW«» 


— 


1032 


C*(P« 


+ 


488 


6V«/ + 


81 


<?eV 


- 64 


ccPe 


+ 


486 


e'd' 


— 


102 


C»(i»/ - 


54 


<?df«P 


- 66 


d* 


— 


81 


a' VP 


— 


2 


c»cfe» - 


136 


<?(P<?f 


- 29 


a» VeP 


+ 


7 


Veep 


+ 


20 


(?d^e + 


150 


<?dtf 


+ 68 


deP 


— 


16 


d^P 


— 


24 


c»d» - 


40 


ecPp 


+ 18 


^P 


+ 


9 


d^P 


+ 


72 






ed^f 


+ 76 


b\?eP 


— 


53 

• 


«y 


— 


64 






cd^(^ 


- 78 


cd?P 


+ 


104 


v<?dp 


+ 


16 


^ 26 




d^ef 


- 27 


cd<?P 


— 


150 


c"«y 


— 


129 


486 




a* b*dp 


+ 24 
- 1 


d}iP 


+ 


117 

48 


cd'ef* 
ed^f 


+ 


108 
72 


8788 
9116 
6880 




v<*p 


+ 1 


d^ff 
df 


+ 


138 


€«■ 


+ 


135 






1 ^ 

- 8 


1 


108 


d*P 


+ 


84 


:!: 90196 




edeP 


+ 46 


v&dp 


— 


82 


d'e'/ 


— 


112 


and see farther 


p. 806. 


c^P 


- 30 


c»«y 


+ 


316 


d»«« 




• • • 







143] TABLES OF THE COVARIANTS M TO W OF THE BINARY QUINTIC. 



293 



T. No. 94. 



y coefficient. 



y coefficient. 



t^b'df' 






a« 6"crf»ey« - 18 


a' b^d'dp - 


75 


a<>6^jd»ey - 


880 


</•* 






cd^f + 150 


<^^P - 


3 


ccPtf* + 


765 


a* hhp 






c«« - 72 


&d^eP - 


108 


d^ef + 


148 


def^ 






dl'ep + 198 


<?df?f + 


308 


^•«' - 


175 


ey 






d»ey - 315 


c»«» + 


112 


6V/» - 


24 


h^(?ef^ 






rf««» + 129 


<?d^P + 


663 


d'dep - 


513 


cd^f 


+ 


1 


h'd'eP + 6 


c«rf»ey - 


783 


cV/ + 


283 


cd^P 


— 


2 


c8rf»/» + 66 


• c»rf«e* - 


306 


i^d^P ~ 


914 


c^P 


+ 


1 


<*d^p - 114 


ccJ'e/' - 


570 


c»rf»ey + 


1986 


^fp 


— 


3 


c>«y + 48 


cd^^ + 


798 


<*d^ - 


280 


d^^P 


+ 


8 


c^d^ep + 9 


^/ + 


216 


t^d'ef + 


527 


d^f 


— 


7 


c«rf»ey - 153 


(i»«» 


252 


c«rf»e» - 


1365 


^ 


+ 


2 


(?d^ + 108 


6^cy» + 


27 


cd^f - 


340 


(^¥p 




• • • 


cd^P - 108 


C'flfe/^ + 


294 


cd^^ + 


700 


b'cep 




• • • 


crf»cy + 396 


c»ey - 


208 


cTe - 


60 


d^P 


— 


1 


cd^^ ~ 240 


c'd^^f - 


93 


6»c*e/« + 


153 


d^P 


+ 


2 


d:'ef - 81 


cV/> - 


570 


^d^P + 


1032 


^r 


— 


1 


d"«» + 63 


c^cfo* - 


28 


(?d^f - 


1098 


h^i?dp 


— 


7 


6V(^» - 18 


c*c?'e/' + 


1116 


c»«* - 


40 


(?^p 


+ 


7 


c»cy> + 6 


c'c^c* + 


224 


i^d^ef - 


1662 


cd'ep 


+ 


30 


c^d^ep + 6 


c»dy - 


369 


c^d^^ + 


1025 


cd^P 


— 


46 


d'd^f + 114 


c«d»c« - 


798 


c'rfy + 


730 


c^f 


+ 


16 


c*«» - 84 


cd?e + 


486 


c»(^e» + 


370 


d^P 


+ 


6 


<?d^p + 42 


d» 


81 


c^cJ'e — 


645 


(Pe'P 


— 


39 


<?d^(?f - 222 


6Vfl/« - 


108 


c^ + 


135 


iP^f 


+ 


53 


c>rf»«* + 144 


i^d^P + 


153 


6^c'(y» - 


486 


A» 


— 


20 


(?d^ef + 54 


c«rfey + 


240 


cV/ + 


144 


h^p 


+ 


6 


c"rf»«» - 42 


c««* + 


88 


d^d^ef + 


1185 


edrf* 


— 


44 


ccPf 


<^d?ef 


474 


c«(ie» - 


60 


e^p 


+ 


20 


cd^^ 


i^d^^ - 


368 


dd^f - 


488 


<?d^p 


+ 


11 


d^e 


d'cPf + 


149 


dd^d - 


880 


c'd'^P + 


105 


a* mp - 5 


c^d^^ + 


578 


dd^e + 


735 


c'd^^/ 


— 


104 


ey* + 5 


c'cJ'e — 


312 


dd^ - 


150 


^^ 


+ 


24 


b'i^P + 7 


0*6? + 


54 


6V/« + 


81 


I cd^eP 


— 


90 


edep - 62 


a« 6 V* 


1 


ddef - 


243 


cd^^f 


+ 


82 


c(^P + 48 


cfe/» + 


28 


dd 


30 


C(?<J» 


— 


16 


(PP + 64 


^P - 


27 


dd^f + 


102 


d^p 


+ 


27 


d'e'P + 6 


fc»C«(5/» - 


11 


dd^d + 


240 


d^^f 


— 


27 


efey - 117 


cd'P - 


68 


c'c?*e — 


190 


d^^ 


+ 


6 


«• +54 


c(iey« - 


12 


c'c?' + 


40 


a* 6V* 




• • • 


• 6Ve/» + 8 


C€*/ + 


108 






l^cdp 


+ 


12 


c'd'P + 29 


d^eP - 


116 






e^P 


— 


12 


i^d^P + 57 


d'e^f + 


234 


i 12 




d}tp 


— 


21 


c>«y - 138 


cfo» 


135 


895 




d^P 


+ 


30 


cfPeP - 238 


6Vci/^ + 


78 


1650 




// 


— 


9 


ceP«»/ + 390 


c»6y + 


12 


6511 

4 *« £^£^£\ 




^<?P 


+ 


12 
69 


cd^ - 72 
rfy« - 194 


i^d^eP + 


513 
735 


11628 




i?dep 


=1=20196 


f?^P 


— 


33 


c^«y + 337 


c»«» 


... 


and Bee farther 


p. 806. 


\ ed^p 


— 


96 


d»^ - 179 


cc^/« + 


274 







5*. y)' 



294 



TAPLW OF THB C0VABIANT8 M TO W OF THK BINABT QUINTIC. [14 









U. No. 29. 








a* b"/* 


. a* J'ePa* - 


22 


a« 6V«i»/» - 108 


a'Wi^P - 


90 


o» VdPP - 


a* 6 V» 


6V/« - 


4 


cW/- 42 


AP^ - 


42 


dV/ - 


6»«i^' 


<!*</«/» + 


36 


<!»«?«« + 298 


<W/ + 


674 


d»«* + 


«!«•/* 


^^P - 


16 


odV" + 242 


<!♦«• 


4 


b*<*dp + 


tPef* 


(?tPP - 


22 


«iV - 294 


c»d'/» + 


394 


<ft?P + 


d^f* 


ifd^P- 


50 


d»/ - 72 


c»dV - 


662 


<?d^P + 


*P 


ifd^f + 


16 


«?«• + 78 


<?<P^/- 


714 


(?dif/ - 


t^Vf* 


<!»«• + 


16 


V'^dp - 6 


<?cPnf - 


498 


<!»«• + 


Vdf* 


c»«?e/» + 


54 


*•«•/» + 62 


c'dV + 


1246 


<*d^P + 


^f* 


e^iff + 


46 


c'dV - 108 


ed'/ + 


224 


<?^^f - 


jvy* 


<!•«?«• - 


60 


fl»«fc»/ - 164 


od»«« - 


516 


<?^P^ + 


cdef* 


t^P - 


6 


c»«» - 24 


d"* + 


48 


afef + 


tPP 


ccfeV - 


70 


cy/* + 63 


6V/» + 


18 


cdV - 


<?«•/» 


«#«♦ + 


56 


cV««/ + 394 


i'ds/' + 


242 


dPf + 


d^P 


rf'e/ + 


18 


cy»«« + 194 


cV/ - 


128 


<P(j» + 


«•/ 


«<•«• - 


14 


edftf - 324 


o'dy - 


324 


6»c»/» - 


6V«/« 


.. a» ft*/* 


« • a 


c»dV - 440 


c'dV/ - 


498 


(*d^ ~ 


c«<?/« - 


1 6««/« 


• • • 


c'd'/ + 78 


c»d(J* + 


136 


&^f + 


c'de»/» + 


2 «?/« - 


1 


«»<?«• + 428 


««V" + 


1078 


e<d»/« - 


c»«y» - 


1 Ay* + 


2 


«*•« - 180 


(A?** + 


206 


cW/ + 


«?«/» + 


6 «y» - 


1 


6P» + 27 


c»dy - 


342 


c«d«» - 


cJP^P - 1 


6 W4^« - 


16 


a> 6V* 


c'd»(j» - 


804 


c'dV + 


«W/ + 1 


4 «V/» + 


16 


ft»«(^« + 14 


(Me + 


506 


c'd»e« - 


ce' 


4 tdPeP + 


82 


e^P - 14 


cd» 


90 


cV/ - 


<?/• - 


4 cefe»/« - 


132 


<P^ - 32 


6V«/-» - 


72 


c^V + 


(^e*/* + 1 


1 c^f + 


50 


d«»/» + 50 


e'tPP + 


78 


eJPe 


cP*?*/ - 1 


««•/• - 


16 


«•/ - 18 


<?d^/ + 


224 


d» 


<;■«• + 


3 df^P - 


14 


bVp - 10 


<?<^ + 


16 


6«c'5r + 


o» 6»^» 


dV/ + 


60 


«W* - 30 


(?tP^ - 


342 


c»d'/» + 


de/* 


d* - 


30 


c»«»/» + 60 


(fdf* - 


220 


C«d9»/ - 


«•/• 


6V/« + 


11 


edVp- 48 


<fd>f + 


106 


c»«« + 


6V«/« 


<>»&/• - 


30 


c«iy + 16 


<fd>^ + 


392 


eepif - 


cd»/' + 


2 c»«y« - 


14 


ed^/ + 88 


cV« - 


222 


c»dV + 


«W/» - 


4 cW/' - 


60 


<!<• - 86 


<?6P + 


40 


cW/ + 


c«y» + 


2 <!»<?«•/• + 


168 


d*ep + 112 


a« Vdp - 


4 


c«V + 


d'«/» - 


6 «!**•/ - 


48 


d**"/ - 204 


t?P + 


4 


c»d«« - 


^^p + 1 


6 c»<C - 


4 


(Pif + 102 


6V/« + 


3 


cV + 


rf**/ - 1 


4 cdV* - 


48 


6»c«</^ + 50 


edrf* + 


24 




^ + 


4 «?«•/ - 


2 


c»d»/» + 46 
<?d?p - 2 
e»««/ - 204 


©«•/» - 


SO 


6'c"<(/*« + 


6 «P«» + 


6 


d»/» + 


16 


ed}«f + 


«!•«•/» - 


6 #/» + 


62 


d»e»/' - 


4 


c'de» - 


«!»</•«/» - 5 


rf-e*/ - 


90 


<?(P^ - 170 


d»«/ - 


36 


efd*/ - 


c'd8'/»+ 8 


2 d»«* + 


39 


«!**• + 308 


if + 


27 


c«d»«» - 


oV/ - 3 


2 iVe/* - 


28 


cV**/ + 42 


i'cdV* - 


104 


c*d»« + 


cd>P + 3 


6 c<d»/» + 


64 


«?/> - 164 


c»^ - 


22 


cW - 


«?«•/• - 3 


<*d^p - 


48 


cdV/ + 674 


c»d»/» - 


60 


6V«/» + 


crf»««/ - 3 


cV/ + 


112 


etP^ - 590 


<fdt?p + 


6 


«•*/ - 


e(i«* +2 


4 c»d'^ + 


82 


dV" - 128 


cV/ + 


102 


c»e» 


dftP - 2 


.8 c»d»«'/ - 


170 


(i>«« + 138 


cdW/ + 


308 


«^d»/ + 


«W/ + 5 


<*d<f - 


104 


W4f* - 70 


cde* - 


234 


c'dV + 














(Cd* + 



rkSe, ^464, db2608, :k7378, d:6878, together :&: 17264: and see farther p. 807. 



143] TABLBS OF THE COVABIANTS M TO W OF THfi BINAftY OUtNtlC. 



iu 



V. No. 95. (•$«, y)». 



» ooeffioient. 



0^ 6»c/* 






<f b*t^' 


^ 


2 


a» 6*orf»«' 


+ 


876 


a« 6Vrf"<^ 


+ 


2800 


d^ 






^f* 


+ 


2 


d^f 


+ 


162 


i^d^rf 


+ 


6624 


«•/* 






V(?P 


— 


16 


rf»fl» 


— 


162 


i^d^^ 


+ 


2052 


«• 6y« 






cdeP 


+ 


32 


a« }^rf^ 


+ 


14 


c^f 


— 


918 


6'c«/» 






c^P 




• • • 


def^ 


— 


6 


ccTe* 


— 


2304 


(Pf 






tPp 


— 


8 


«•/• 


— 


8 


d»« 


+ 


486 


«fay* 






«?«•/» 


+ 


80 


b'c'ef' 


— 


50 


c'dp 




• • • 


«•/* 






d^P 


— 


160 


ed"/' 


+ 


90 


h'(?i?P 


+ 


504 


V<?df* 


— 


2 


</• 


+ 


72 


cd^P 


— 


120 


i^d^p 


— 


576 


<(•«»/< 


+ 


2 


ve«p 


+ 


84 


c^r 


+ 


60 


i^d^f 


— 


2288 


«?«/* 


+ 


10 


^dfp 


— 


104 


^rf^ 


— 


280 


i^4^ 


+ 


1172 


ed^P 


— 


16 


ifd^P 


— 


160 


d^^P 


+ 


300 


<*d*p 


— 


124 


ofP 


+ 


6 


«;•««/• 


+ 


60 


d^f 


+ 


216 


c»rf»«y 


+ 


4336 


itf* 


— 


6 


cefep 


+ 


320 


^ 


— 


216 


dd^f^ 


— 


2540 


^^f* 


+ 


12 


et^P 


+ 


80 


h'i^df^ 


— 


160 


d'd^ef 


— 


1912 


dV/» 


— 


10 


odff 


— 


496 


&^P 


— 


80 


c*d*«» 


+ 


2100 


d^f 


+ 


6 


eg 


+ 


252 


^d^P 


+ 


1280 


^d?f 


+ 


240 


«• 


— 


2 


dPp 


— 


72 


<?d^P 




• a • 


&d^^ 


— 


1560 


a* 4«^ 




• • • 


dVp 


— 


420 


c»</" 


— 


312 


&d^€ 


+ 


810 


Vedf* 


+ 


4 


ePe*/ 


+ 


860 


cd^P 


— 


440 


ecP^ 


— 


162 


c^f* 


— 


4 


dV 


— 


404 


cd^e^P 


— • 


2160 


a» 6y» 


— 


4 


d^ef* 


— 


10 


b'^dp 


+ 


96 


edS^f 


+ 


1740 


h^CBP 


-1- 


22 


«&•/» 


+ 


16 


cV/' 


— 


120 


cd^ 


— 


216 


d'P 


~ 


26 


«•/• 


— 


6 


i?d?^P 


— 


560 


d^tp 


+ 


2344 


dd'P 


+ 


76 


WP 


+ 


6 


«?dgp 


+ 


160 


d^ef 




3240 


^P 




• • • 


^def* 


— 


26 


«»«•/ 


+ 


304 


rf»«» 


+ 


1244 


}^i?dp 


+ 


124 


«•«•/• 


+ 


8 


<?d^P 


+ 


280 


WP 


+ 


72 


f^^P 


+ 


368 


ed^f* 


+ 


32 


^dfgp 


+ 


1440 


(MeP 


— 


240 


cdFep 


— 


688 


«?«•/» 


— 


116 


<?<Pe*f 


— 


900 


&i?P 


+ 


940 


cd^P 


— 


192 


tdi^P 


+ 


180 


e'dtf 


- 


376 


f?d^P 




• • « 


^f 




• • • 


«*•/ 


— 


78 


ed^eP 


— 


1296 


&d^^P 


— 


1320 


d^P 


+ 


400 


<ft./» 


+ 


24 


ed'g/ 


+ 


80 


&df^f 


— 


2640 


d^^P 


+ 


984 


<?«!•/» 


— 


20 


cd^g 


+ 


832 


fl»^ 


+ 


908 


d^^f 


— 


2160 


««•«•/ 


— 


44 


«r/» 


+ 


432 


i^dSp 


+ 


600 


d^ 


+ 


1080 


dg 


+ 


34 


dfgf 


— 


72 


&d^^f 


+ 


3360 


h^d'p 


— 


60 


6»«V* 


— 


30 


dP^ 


— 


240 


(?d^f^ 


— 


168 


^dep 


— 


480 


^d^f* 


+ 


4 


Wp 


— 


36 


ed^P 


— 


1656 


&^p 


— 


1580 


ifd^fp 


+ 


240 


ifdtp 


+ 


288 


fid^^f 


+ 


3408 


i?d?P 


+ 


40 


«•«*/» 


— 


130 


(fgp 


— 


56 


ed"^ 


— 


3480 


f^d^^P 


+ 


2040 


^^»P 


— 


160 


<^d}p 


— 


140 


d?ef 


— 


1008 


i^dt^f 


+ 


2910 


(?(Pt?P 


— 


280 


(fdP^P 


— 


480 


d^i^ 


+ 


1224 


c»^ 


— 


810 


^d^f 


+ 


332 


iNitff 


+ 


420 


h'ifep 


— 


144 


6d^€f^ 


— 


3420 


«»«' 


— 


54 


«?•<*• 


— 


276 


&d^P 


+ 


108 


cdFi^f 


+ 


4800 


efiP 


+ 


24 


ed^«P 


+ 


420 


^d^P 


— 


768 


ed"^ 


— 


3510 


td^P 


+ 


360 


edfgf 


— 


1120 


&^f 


— 


700 


d^p 


— 


1516 


«?«♦/ 


— 


320 


&d'f 


+ 


1112 


d^eP 


+ 


900 


(P^f 


+ 


2156 


edV 


+ 


38 


cW/» 


— 


144 


f^d^f 


+ 


8160 


d^ 


— 


430 


dfiP 


— 


108 


i?*^f 


+ 


1620 


C*(fo» 


— 


2148 


W^ 


+ 


336 


#«•/ 


+ 


96 


(MV 


— 


1620 


&6^P 


+ 


912 


d'd^P 


— 


40 


#«• 


— 


12 


ed'ef 


— 


864 


^d^^f 


— 


15060 


f^d^p 


+ 


2640 



For the Nomerioftl Yerifioations see p. 806. 



296 TABLES OP THE COVAMANTS M TO W OF THE BINABT QUINTIC. [143 



V. No. 95 (continued). 



X coefficient. 



a' i'c'e*/ 


+ 


1840 


a" Vd^p 


+ 


• 

184 


a» 6*c»«/« 


- 594 


«•<?«/» 


— 


1280 


d^P 


■ 


108 


ed^p 


- 10296 


<?^i?f 


— 


13360 


«v 




• • • 


(?d^f 


+ 10080 


c»<W 


+ 


3200 


v<?p 


+ 


18 


(?^ 


+ 900 


<?dfP 


+ 


7312 


i?deP 


+ 


264 


&d?^ 


+ 19440 


^d^ff 


— 


2360 


<?^P 


+ 


756 


C«(?«» 


- 8800 


<?dP^ 


+ 


3840 


c^P 


— 


368 


f^iPf 


- 9160 


cd^^ 


— 


5344 


cd^^P 


— 


732 


i^d^ 


11900 


ed*(? 


+ 


2800 


od^f 


+ 


540 


c*d»« 


+ 13900 


rfy 


+ 


1956 


ce« 




• • • 


c»rf» 


- 3150 


iPt? 


— 


1680 


d^p 


— 


1172 


h^&dtp 


+ 3564 


6Vd/' 


— 


36 


df^f 


+ 


2520 


c»«y 


- 1350 


«•«•/» 


— 


1296 


rf*** 


— 


1350 


c»dV 


- 9540 


<?dPeP 


+ 


1668 


6'c*e/» 


— 


144 


i^d^ 


- 750 


<*dtf/ 


— 


1312 


^d?p 


+ 


376 


ed^f 


+ 4260 


«•«• 


— 


2060 


<?d4?p 


— 


1440 


dd^i? 


+ 10800 


cV/* 


— 


8020 


(f^/ 


— 


1530 


c«d»c 


- 9100 


cy«»/ 


+ 


15220 


cW«/« 


+ 


6360 


<*d? 


+ 2000 


c«dV 


+ 


1180 


<»d^<?/ 


— 


6000 


b^c'P 


- 486 


<»eP^ 


+ 


3712 


i?d^ 


+ 


1350 


^'def 


+ 1620 


ifd'tf 


-- 


8540 


cd?P 


+ 


2344 


c^V 


+ 450 


^d^f 


— 


2952 


ed*^/ 


— 


9260 


&d^f 


- 720 


<!•<?«» 




• • • 


edfif 


+ 


7200 


&d^ 


- 2250 


cd*a 


+ 


3330 


<??/■ 


+ 


1720 


d'd'e 


+ 1800 


<P 


— 


810 


<?«• 


— 


1900 


dd" 


- 400 


wp 




• • • 


h\*d{p 


— 


168 






e'd'p 


— 


576 


«•«•/• 


+ 


648 






e^f 


+ 


1824 


t^tNp 


~ 


6420 






i?d?p 


+ 


3792 


cW/" 


+ 


9360 






c»d'«y 


— 


5808 


cV 


+ 


450 






tfd^ 


+ 


3240 


«•#/» 


— 


10100 






<fd^/ 


— 


4768 


&dfff 


+ 


19920 






<*d?^ 


— 


6240 


<?dW 


— 


10300 






(^dPf 


+ 


2608 


cWe/ 


+ 


4920 






c*^<? 


+ 


12440 


cW 


— 


10100 






<?(Pe 


— 


8160 


cdff 


— 


3440 






<^cP 


+ 


1620 


cdV 


+ 


7100 






b'<»eP 


+ 


162 


d»« 


— 


750 






(i'd'P 


~ 


702 


wp 


+ 


36 






c»rf«»/ 


— 


90 


ifdip 


+ 


2988 






c««« 


— 


1290 


&i?f 


— 


2880 






c'dV 


+ 


1920 


<fdfp 


+ 


14688 






c'dV 


+ 


3640 


<fd^i?/ 


— 


22740 






<fd'/ 


— 


796 


<*d^ 


+ 


600 






&d^ 


— 


5340 


<*d^ef 


— 


16520 






fl»d»« 


+ 


3100 


<!«(?«• 


+ 


23300 






<j^d» 


— 


600 


edff 


+ 


8760 






a* Vip 


+ 


18 


«•<?«' 


— 


5200 






Vcdf< 


— 


36 


<?d!t 


— 


5400 






m^P 


— 


180 


<5rf» 


+ 


1500 







143j TABLES OF THE CO VARIANTS M TO W OF THE BINARY QUINTIC. 



297 



V. No. 95 (continued). 



y coefficient. 



«• fdf* 






a» V'<?deP 


„«• 


116 


a« 6 V/ 




• • • 


a^ 6Vrf*ey 


+ 


7312 


_. **/' 






e'^P 


+ 


80 


6Vc/* 


— 


20 


i^d^ 


+ 


2344 


«• *'«/• 






cd'P 


+ 


240 


<^<Pf' 


— 


280 


i^d^P 


— 


124 


dtf* 






eePe'P 


— 


160 


<?d^p 


+ 


80 


c*^ey 


— 


8020 


^r 






ed^P 


— 


120 


i?e^p 


+ 


300 


c*d^ 


— 


10100 


*Ve/» 






«•/ 


+ 


76 


cd?ep 


+ 


160 


C»(f»«/ 


+ 


3792 


«P/» 


— 


2 


tfep 


— 


120 


edJ'^P 




• • • 


c>rf»«» 


+ 


14648 


«fcy* 


+ 


4 


d^e^P 


— 


80 


cd^f 


— 


192 


<?d^f 


— 


702 


a^P 


— 


2 


(P^f 


+ 


368 


c^ 


— 


108 


&d?^ 


— 


10296 


dfef* 


+ 


6 


dg 


— 


180 


<PP 


— 


56 


cd^e 


+ 


3564 


df<?P 


— 


16 


V<*ep 


+ 


24 


d}^P 


+ 


940 


J" 


— 


486 


d^P 


+ 


14 


<^cPp 


— 


160 


d^e'f 


— 


1580 


a^ 6V' 


+ 


6 


* ^f 


— 


4 


<*d^P 


+ 


320 


d"^ 


+ 


756 


deP 


— 


78 


<»* iy« 




• • • 


<?eP 


— 


280 


6Vrf/* 


+ 


360 


^P 


+ 


72 


*•«/» 




• • • 


^d^eP 


— 


560 


cV/» 


— 


420 


h^(?eP 


— 


44 


d»/' 


+ 


2 


ff^^P 


+ 


1280 


&d^eP 


+ 


1440 


cdPP 


+ 


332 


d^P 


— 


4 


i?d<ff 


— 


688 


(?d^P 


— 


2160 


cd^P 


— 


496 


«y 


+ 


2 


i?g 


+ 


184 


<^^f 


+ 


984 


c^P 


+ 


216 


b'c'dp 


+ 


10 


ctPp 


+ 


288 


c^d^p 


— 


480 


d^eP 


+ 


304 


e^P 


— 


10 


cd*^P 


— 


240 


&d^eP 


— 


1320 


d'i^P 


— 


312 


e<PeP 


— 


26 


ccPey 


— 


480 


(?d^e^f 


+ 


2040 


d^f 




• • • 


ed^P 


+ 


32 


«?«• 


+ 


264 


(?d^ 


— 


732 


^ 




• • • 


c^P 


— 


6 


dPtP 


— 


144 


cd^ep 


— 


768 


b*(^dp 


— 


320 


d'P 


— 


30 


d^<?f 


+ 


336 


cd*^f 


+ 


2640 


<?^P 


+ 


860 


d*^P 


+ 


84 


d*^ 


— 


144 


cd^^ 


— 


1440 


i^d^eP 


— 


960 


df(^P 


— 


50 


b'i^dp 


+ 


24 


d^P 


+ 


504 


&d^P 


+ 


1740 


*•/ 


— 


22 


<f^P 


— 


72 


d^^f 


— 


1296 


<?^f 


— 


2160 


«» 


+ 


18 


d'cPeP 


+ 


280 


d^e'' 


+ 


648 


cd'P 


+ 


420 


wp 


— 


6 


(fd^P 


— 


440 


b'(^P 


— 


108 


cd^i^P 


— 


2640 


&d»P 


+ 


32 


C*«»/ 


+ 


400 


d'dep 


— 


1296 


cd^^f 


+ 


2910 


e^P 


— 


8 


&d*P 


— 


140 


i?^P 


+ 


2344 


cd(^ 


+ 


540 


<?<pp 


+ 


4 


^df^P 




• • • 


d'd^P 


+ 


420 


d^eP 


— 


700 


<M}i?P 


— 


104 


c»d«ey 


+ 


40 


i^d'^P 


+ 


600 


d^f 


+ 


1840 


<?dt^P 


+ 


90 


c»<fe« 


— 


368 


<^d/f 


— 


3420 


d^ 


— 


1530 


c"**/ 


— 


26 


c"d»«/» 


+ 


108 


d"^ 


— 


1172 


h^^P 


+ 


96 


cd*eP 


+ 


96 


<?d'^f 


— 


40 


&d^ep 


+ 


900 


d'dep 


+ 


80 


edf^P 


— 


160 


c»d»«» 


+ 


376 


i^d^f 


— 


1280 


f^^P 


— 


3240 


«?«•/ 


+ 


124 


'<ap 




• • • 


^d!"^ 


+ 


6360 


&dp 


— 


1120 


C(*^ 


— 


36 


c^^f 


— 


36 


i?d^P 


— 


576 


&d^P 


+ 


3360 


dFp 


— 


36 


cd?<^ 


— 


168 


i^de^f 


+ 


1668 


ifde^f 


+ 


4800 


df^P 


+ 


72 


dPef 




• • • 


c«rfV 


— 


6420 


c>«« 


+ 


2520 


d>t^f 


— 


60 


d?t? 


+ 


36 


cdef 


.— 


576 


d'dep 


+ 


8160 


__ <?«• 


+ 


18 


o« 6»<^» 


+ 


6 


cd^e^ 


+ 


2988 


^d^f 


— 


13360 


**• h'ep 




• • • 


«y' 


— 


6 


d^f 


+ 


162 


i?d^ 


— 


6000 


Vcdp 


— 


16 


6V/» 


— 


10 


d»(5" 


— 


594 


cdp 


— 


2288 


c^P 


+ 


16 


cdrf* 


+ 


180 


b^<FeP 


+ 


432 


cdt^f 


— 


1312 


d?ep 


+ 


8 


c^P 


— 


160 


d'd^P 


— 


144 


cd*^ 


+ 


9360 


d^P 




• • • 


tpp 


— 


130 


(^d^P 


— 


1656 


def 


+ 


1824 


«•/' 


— 


8 


d»ey> 


+ 


60 


cV/ 


— 


1516 


d^ 


— 


2880 


6V/» 


+ 


12 


dt*P 


+ 


60 


&d^eP 


+ 


912 


h^d'eP 


— 


72 



For the Nmnerical Yerifioatioxis see p. 808. 



C. II. 



38 



298 



TABLES OF THE OOVARIANTS M TO W OF THE BINABT QUINTIC. [143 



V. No. 95 (concluded). 



y coefficient. 



a^ b^d'cPf* 


+ 


1620 


a« h^P 


^^ 


276 


a« 6*c«<fo* 


+ 7100 


&der 


+ 


3408 


d^t^P 


+ 


908 


i*d^ef 


+ 12440 


&^f 


+ 


2156 


d}^f 


— 


810 


c-d*** 


- 5200 


c'cPep 


— 


15060 


d^ 




... 


<N^f 


- 5340 


c^d^^f 


— 


2360 


6»cy* 


— 


12 


c'tPe' 


- 11900 


c*cfe» 


— 


9260 


<^d^' 


+ 


832 


&d?t 


+ 10800 


i?d'P 


+ 


4336 


&i?P 


■♦• 


1244 


c«d» 


- 2250 


i?d^^f 


+ 


15220 


i^d^P 


+ 


1112 


h'd'ep 


+ 486 


(*<Pt^ 


+ 


19920 


c'd't^P 


-• 


168 


i?d}p 


+ 810 


d'd^ef 


— 


5808 


<?de'f 


— 


3510 


i^d^f 


+ 3330 


<?iJ^f? 


— 


22740 


c«e« 


— 


1350 


c»«* 


- 750 


cd^f 


— 


90 


cd^fp 


— 


2148 


ed^tf 


- 8160 


Cd^i? 


+ 


10080 


cd^^f 


+ 


3200 


ed^i? 


- 5400 


dl'e 


— 


1350 


cd'tfi 


+ 


1350 


d^d^f 


+ 3100 


h^edp 


— 


864 


d^P 


+ 


1172 


i^d^t? 


+ 13900 


e^P 


— 


1008 


d»«»/ 


— 


2060 


<*d^e 


- 9100 


<^d?ep 


+ 


6624 


c^«^ 


+ 


450 


<jW 


+ 1800 


i^d^f 


— 


5344 


h'&eP 


— 


240 


h^i^^dp 


- 162 


i^i? 


+ 


1720 


<^d}p 


— 


1620 


e^^i^f 


- 810 


&d>P 


— 


1912 


c'd^P 


— 


3480 


(^d}ef 


+ 1620 


&^i?f 


+ 


3712 


<?*</• 


— 


430 


&df? 


+ 1500 


c»cr»«^ 


+ 


4920 


<*d^ep 


+ 


2800 


i^d^f 


- 600 


i^d^tf 


— 


4768 


i?d^i^f 


+ 


3840 


d^d^t? 


- 3150 


d'd^ 


— 


16520 


&d^ 


+ 


7200 


dd^e 


+ 2000 


&d^f 


+ 


1920 


<?d^P 


— 


2540 


i^d? 


- 400 


C«(?«» 


+ 


19440 


i^d^i^f 


+ 


1180 






c»d"« 


— 


9540 


c»cP«* 


— 


10300 






cdy" 


+ 


1620 


Cd^if 


+ 


3240 






6V/» 


+ 


162 


cd»«» 


+ 


600 






^dep 


— 


918 


d^f 


— 


1290 






i»^f 


+ 


1956 


d?i? 


+ 


900 






ed^p 


+ 


240 


V't^dp 


+ 


876 






(?d^^f 


— 


2952 


c-eV 


+ 


1224 






edi^ 


— 


3440 


<^d^^P 


+ 


2052 






d^d^ef 


+ 


2608 


&di?f 


+ 


2800 






i*d^^ 


+ 


8760 


&^ 


— 


1900 






&d^f 


— 


796 


^d^P 


+ 


2100 






&d^i? 


— 


9160 


d^d^i^f 


— 


8540 






c^d'e 


+ 


4260 


c*cf«* 


— 


10100 






c»rf» 


— 


720 


(?d^ef 


— 


6240 






a' bp 


— 


2 


i^d'i? 


+ 


23300 






l/cef* 


+ 


34 


i?d!f 


+ 


3640 






d"/* 


— 


54 


e^i? 


— 


8800 






d^P 


+ 


252 


c<Pe 


— 


750 






eP 


— 


216 


d}' 


+ 


450 






h^c'dp 


+ 


38 


WP 


— 


162 






i?^P 


— 


404 


t?deP 


— 


2304 






cd'eP 


— 


376 


t?^f 


— 


1680 






cd^p 


— 


216 


i^d^P 


— 


1560 






c^f 


+ 


1080 


&d^^f 




« • • 







143] TABLES OF THE COVARIANTS M TO W OF THE BINARY QUINTIC. 



299 



W, 29 a. 



a»6«/» 


• « • 


o^ h^c^P 


- 1 


5 a» 6Vdey* - 


90 


a» 6V(£«fiy 


+ 1320 


a»6'e/« 


• • • 


d^^P 


+ 1< 


cV/» + 


30 


<?d^ 


- 260 


b'cdp 


« « • 


(t^P 


- 3. 


5 cd'ep - 


210 


f?d?p 


+ 60 


c^r 


• • • 


iP^P 


+ 4( 


ctPe'P + 


120 


d'd^^P 


- 500 


cPeP 


• • • 


d^^P 


- 1< 


cd'^p + 


360 


d'd'ep 


+ 2235 


<&»/• 


• • • 


dA?f 


- li 


cd^P - 


420 


(^d*d'f 


- 1995 


«•/• 


• • • 


^ 


+ 


5 cc»/ + 


130 


i?d^^ 


+ 370 


a' If dp 


• • • 


h^i^P 


— 


1 d^P - 


5 


cd^ep 


+ 360 


^r 


• • • 


c^deP 


+ 1. 


5 rfV/* + 


195 


cd?^P 


- 1320 


1fi»p 


• • « 


c'i^P 


- K 


a d'eY* - 


315 


Cf^^f 


+ 1110 


edeP 


• • • 


&d?P 


• • 


d^d'P + 


40 


cd^^ 


- 210 


e^P 


• • • 


c'cPe'P 


- 9< 


rf»ey + 


165 


d^P 


- 81 


tPp 


• • • 


(^dk^P 


+ 12< 


d^^ - 


75 


d^^P 


+ 270 


«?«•/• 


• • • 


i?^P 


- 4< 


6V€/« - 


10 


d^^f 


- 225 


dt/'P 


• • • 


i?d'tP 


+ 61 


c*rf»/« - 


60 


cfe« 


+ 45 


«•/' 


• • « 


f^d^t^p 


+ 31 


D i^d^p + 


210 


a* U'eP 




veep 


• ■ • 


f?d^^P 


- 18 


c*«y* - 


110 


h^cdp 




tftpp 


• • • 


(?dep 


+ 12 


i?d^ep 


« • • 


ceP 




<?di?P 


• • ■ 


<?i?f 


- 2 


f^d'^p 4 


60 


d^eP 




<*e*P 


• ■ • 


cd^P 


- 1 


5 i?d&p - 


360 


d^P 




ed^eP 


• • • 


c^^p' 


- 11 


<?^p + 


240 


^r 




cd'^P 


• • • 


cd^eP 


+ 26 


5 c«rfy» + 


30 


h'i^p 


- 10 


cd^p 


• • • 


cd'^P 


- 20 


i?d^^p - 


210 


i?deP 


+ 90 


cey 


• • • 


cd^/f 


+ 6 


5 i^^ep - 


180 


d'^P 


- 60 


d^P + 


1 


cde'^ 


- 1 


&d}^p + 


1140 


cd^P 


- 120 


d^^P - 


5 


d?eP 


+ 4 


5 i^d^f - 


870 


cd^i?P 


+ 90 


dPe*p + 


10 


d^e'P 


- 10 


i?^^ + 


130 


cdt^P 


• • « 


ePe*/' - 


10 


d^i^P 


+ 8 


1 cd^ep + 


310 


c^P 


• • • 


<fc»/» + 


5 


d^e'f 


- 3 


cd^d'P - 


240 


d'ep 


+ 110 


«"/ - 


1 


rf»«» 


+ 


5 cd^i^p - 


390 


d^^P 


- 50 


cfVcP 


• • • 


a» h^p 




ccPey + 


280 


d^^P 


- 240 


dtp 


• • « 


h'ceP 




ccPe* + 


30 


d^P 


+ 280 


ey 


• ■ • 


d?p 




d^P - 


180 


^f 


- 90 


V<?eP 


• • • 


d^P 




(fey» + 


300 


h^i^eP 


+ 35 


cd'P 


• • • 


eP 




dJ'^p - 


120 


i?d^P 


- 30 


ed^P 


• • • 


h^i^dp 




d^^f + 


30 


(^de'P 


- 120 


ofP 


• • • 


i?^P 




rf*e» 


30 


<?tP 


+ 50 


<?«/• 


« • « 


cd'eP 




\P(^dp + 


15 


<?d^eP 


- 60 


dPt»p 


• • • 


cd^P 




&^P + 


5 


i?d^i?P 


« • • 


d^P 


• • • 


cd'P 




&d^eP - 


30 


i^d^P 


+ 360 


^r 


• • • 


dp 


+ i 


&d^p - 


270 


i^dp 


- 210 


\f<*dp 


• • • 


d^^P 


- 4 


&^p + 


196 


cd^P 


+ 270 


c'e'/' 


■ • • 


d^^P 


+ 6 


c^d^p 


« • • 


cd^^P 


+ 575 


<»d*eP 


• • • 


d^P 


- 4 


c^d^^p + 


225 


ccPtp 


- 1700 


<?d^P 


• • • 


^P 


+ 1 


(^d^tp + 


615 


cd^^P 


+ 480 


i?^P 


• • • 


h'd'P 


+ 


5 i^d^P - 


660 


cd^f 


+ 670 


cd*P - 


15 


c'deP 


- 6 


c*ey + 


45 


ce" 


- 315 


cd?<?P + 


60 


c't^P 


+ 4 


i^d'ep - 


120 


d^eP 


- 685 


ccPey* - 


90 


i?^P 


+ 9 


&d^^p - 


220 


d^^P 


+ 540 


««*?•/» + 


60 


c'd'e'p 


• 


i?d?^P - 


980 


d^i^P 


+ 1515 



For the Numerioal Verifications see p. 809. 



38—2 



300 



TABLES OF THE COVARIANTS M TO W OF THE BIXART QUINTIC. 



[143 



W, 29 A (contmued). 



a* 6»<i'«y 




2080 


a* 6^C(^^ 


^^^ 


615 


a»6»rf»/» 


__ 


196 


a^6*(?e« 


- 3880 


9f 


+ 


705 


d}^ep 


— 


945 


dV/« 


— 


660 


h^dep 


- 300 


b^(*df* 


+ 


110 


d^f 


+ 


900 


dP^r 


+ 


1840 


i^d^p 


+ 500 


<^^r 


— 


195 


d"«» 


+ 


45 


<?«•/» 


— 


1040 


d'd^P 


+ 3810 


d'iPef^ 


+ 


210 


h^d^ep 


+ 


180 


d^f 


— 


180 


i?^p 


- 3710 


i^d^P 


— 


575 


edp 


— 


60 


«»»» 


+ 


216 


ed^tp 


- 14040 


i^i^P 


+ 


660 


ed^p 


— 


1420 


h*<^df* 


— 


265 


&d^^P 


+ 16120 


c'd'f^ 


— 


225 


cV/» 


+ 


25 


<?♦«•/• 


+ 


315 


&d^P 


- 540 


i^J^^P 


+ 


1350 


i^d^ep 


+ 


780 


<*(Pef* 


+ 


180 


c»ey 


+ 600 


^d^^P 




• • • 


i^d^^p 


+ 


5760 


<»d^P 


+ 


1700 


d'd^P 


+ 7020 


i^d^P 


— 


1440 


i*d^P 


— 


2945 


(!»«•/• 


— 


1840 


i^d't^P 


• • • 


c»ey 


— 


75 


(^ey 


+ 


1390 


<?d>P 


— 


615 


d'd^e^P 


- 1950 


(?d^ep 


— 


1965 


&d^P 




... 


t?<Pi?/^ 


— 


1350 


i^d^^f 


- 17670 


<?d^^P 


+ 


6000 


&d^^P 


— 


7020 


^d?/f* 




• • • 


C*(fo» 


+ 4170 


<?d^^P 


^ 


7050 


&d^tP 


— 


180 


c'tfc'/* 


+ 


1560 


&d^tp 


+ 480 


i?d^^f 


+ 


3000 


&d}iff 


— 


1275 


<?*/ 


+ 


135 


&^^P 


- 31040 


i?d^ 


+ 


265 


&d^ 


— 


1110 


cd^ef* 


+ 


2210 


ed^^f 


+ 45180 


cd^P 


+ 


1420 


c^d^eP 


+ 


3120 


cd}<^P 


— 


4100 


<*d^d 


- 3160 


cd'^P 


— . 


3810 


d'd^d'P 


+ 


3900 


ccPfp 


+ 


6000 


f?d^P 


- 140 


cd^i^P 


+ 


2310 


(^cy 


+ 


1240 


ccP^f 


— 


4880 


i^d'^P 


+ 18000 


cd^^f 


+ 


1795 


cV«' 


+ 


3155 


oM 


+ 


990 


i^d^^f 


- 12180 


cd^^ 


_^ 


1800 


i?d^P 


— 


515 


<r/« 


— 


25 


<?d^i^ 


- 13430 


^eP 


+ 


240 


i*d^P 


— 


2920 


d-^/* 


+ 


3710 


cd^eP 


- 7200 


d^d'P 


+ 


30 


&d^e^f 


— 


940 


d««y* 


— 


10755 


cd^^f 


- 120 


d^^f 


__ 


870 


c«d»e« 


— 


4300 


d^f 


+ 


9875 


cd?^ 


+ 9960 


d^^ 


+ 


615 


^d^ep 


+ 


675 


<?«• 


— 


2845 


d'P 


+ 1890 


h^eP 


_ 


45 


ed^^f 


+ 


510 


6V/« 


+ 


100 


rf^v/ 


- 540 


i^dtp 


_ 


310 


i?d'^ 


+ 


2940 


&def* 


+ 


240 


d»e* 


- 1710 


i?^P 


+ 


685 


cd>P 




... 


<!•«•/* 


— 


540 


b'ddp 


- 360 


&d^P 


+ 


120 


cd'^'e'f 


— 


135 


<*d}f* 


+ 


220 


de'P 


- 240 


i^d'e^P 


+ 


1965 


cd^e' 


— 


990 


<^d>^f* 


— 


6000 


dd^P 


+ 5840 


&deP 


— 


2210 


rfiy 




• • • 


e*d6*P 


+ 


4100 


dd^P 


- 6560 


&^P 


— 


960 


c^V 


+ 


135 


c***/* 


+ 


1340 


ddp 


+ 8460 


e'd'ep 




• • « 


c^h'dp 




• • • 


ifd^P 


4- 


11700 


dd^P 


- 3120 


c'd^d^P 


-— 


11700 


^P 




• • • 


t^dP^P 




• • • 


d^dp 


- 480 


i^d^^P 


+ 


15435 


h^i?P 


+ 


10 


<*d^^P 


— 


15240 


dd^dP 


- 25880 


i^d^f 


— 


2760 


CileP 


— 


60 


edgf 


+ 


6960 


dddf 


- 1820 


d"^ 


+ 


555 


c^P 


+ 


40 


&i? 


— 


1620 


dd 


- 3620 


<?d^P 


— 


780 


d^P 


+ 


40 


<»d*/* 


— 


5760 


dd^eP 


+ 49680 


&d^^P 


+ 


14040 


d^^P 


— 


30 


c»d'«y» 


— 


16120 


dd'dp 


« « • 


i?d^tp 


— 


10625 


de'p 




• • • 


c'd'e'p 


+ 


26700 


dd^df 


+ 17520 


i^dl'^f 


— 


3220 


^P 




• • « 


<?d'<?/ 


— 


5240 


dd'd 


+ 13500 


&d}^ 


^ 


570 


h^&eP 


— 


40 


c'rf"*' 


— 


1640 


dd^P 


- 120 


(^deP 


— 


5840 


i^d'P 


+ 


180 


ed?ef* 


+ 


6560 


dd^dp 


- 32280 


i?d^^P 


— 


540 


f^d^P 


— 


360 


od*^r 


+ 


7240 


dd^df 


- 46880 


&d^^f 


+ 


5550 


c»«y* 


+ 


240 


cd^ff 


— 


24240 


dd^d 


- 30040 


c*rfV 


+ 


1285 


cd^eP 


+ 


360 


ed^d 


+ 


11420 


dd^eP 


+ 12860 


cd^P 


+ 


990 


cd^^P 


— 


360 


d^f* 


— 


980 


dd^df 


+ 32000 


cd^^p 


+ 


3150 


cd^P 




• • • 


(P^P 


— 


3420 


dd'd 


+ 46160 


cdH^f 


— 


3600 


c^P 




• • • 


d?^f 


+ 


8100 


dd'^P 


- 2700 



143] TABLES OP THE COVARIANTS M TO W OF THE BINARY QUINTIC. 



301 



W, 29 A (continued). 



«» b^<NPi?f 


_ 


8820 


a» l^cd^^p 


+ 


1440 


a» h^cd^d 


^^ 


18750 


a« h^dd?d 


+ 


243000 


C>(i»«* 


— 


34620 


cd^d'P 


— 


1560 


d^ep 


+ 


14115 


dd'^ef 


+ 


2340 


cd^tf 


+ 


1080 


ed^P 




• • • 


d^df 


— 


23790 


dd^d 


— 


89550 


cd^t? 


+ 


12060 


c^f 




• • • 


d?d 


+ 


8175 


cd^f 


+ 


270 


d^f 




• • • 


d^tp 


+ 


960 


Wdp 


+ 


1320 


cd>^d 


+ 


15120 


d^e" 


— 


1620 


d^^P 


— 


1340 


ddp 


— 


30 


d^e 


— 


810 


b^c'T 


+ 


81 


d^f^P 


— 


2440 


dd^ep 


+ 


540 


b'd'ep 


+ 


945 


edep 


— 


990 


d^df 


+ 


4320 


dddp 


— 


7240 


dd^p 


— 


675 


f?^P 


+ 


980 


dt^ 


— 


1620 


ddp 


— 


20390 


dddp 


+ 


7200 


^d^P 


+ 


515 


\^&P 


— 


81 


dd^P 


— 


3900 


ddp 


— 


14115 


f?d?^P 


+ 


UO 


d'dep 


+ 


390 


dd^dp 


+ 


31040 


dd^eP 


— 


12860 


^d^p 


— 


195 


d'dp 


— 


1515 


dd^dp 


+ 


32370 


dd^dp 


+ 


8220 


c»ey 


— 


5575 


dd^P 


+ 


980 


dddf 


+ 


38820 


dddf 


+ 


150 


^d^tP 


+ 


120 


dd^e^P 


+ 


7050 


dd 


+ 


9310 


dd 


+ 


6155 


ed^^p 


— 


800 


ddtP 


— 


6000 


dd^ep 


— 


49680 


dd^P 


— 


480 


dd^i^f 


+ 


22600 


d^P 


+ 


2440 


dd*dp 




• • • 


dd'dp 


+ 


26700 


ddd 


+ 


7240 


d^ep 


— 


15435 


dd^df 


— 


91260 


dd^df 


+ 


63960 


f^d^p 




• • ■ 


ddPdP 


+ 


15240 


dd^d 


— 


50550 


dd^d 


— 


6660 


f^d^^p 


— 


1260 


dd^dp 




• • • 


dd^P 


+ 


800 


dd^ep 




• • • 


d^d^/f 


— 


42330 


dddf 


— 


6480 


dd^dp 


+ 


81840 


dd^df 


— 


180600 


c«^e« 


— 


34340 


dd 


+ 


1215 


dd^df 


+ 


360 


dd^d 


— 


71610 


f?d'tp 


+ 


480 


ed^P 


+ 


2945 


dd*d 


+ 


101450 


dd^P 


— 


4755 


ffd^^f 


+ 


48360 


cd^dp 


+ 


540 


dd^ep 


— 


8220 


dd'df 


+ 


141240 


C»(i»C» 


+ 


73828 


ed^dp 


— 


795 


dd^df 


— 


58080 


dd^d 


+ 


219730 


c*(iy» 


+ 


105 


cd^df 


— 


4180 


dd^d 


— 


34300 


dd^ef 


— 


45130 


c^d^f 


— 


30265 


cd}d 


+ 


4185 


cd}p 


— 


7590 


dd^d 


— 


240975 


C*(f<?* 


— 


92290 


d^ep 


— 


8460 


cd^df 


+ 


41640 


dd^f 


+ 


5580 


ed>^ef 


+ 


9540 


d^dp 


+ 


20390 


cd^d 


— 


4650 


dd'^d 


+ 


128490 


c»(i»e» 


+ 


69220 


dPdf 


— 


16194 


d}^ef 


— 


5580 


dd^e 


— 


34155 


c»(P/ 


— 


1215 


d^d 


+ 


3765 


d}^d 


+ 


1980 


cd"* 


+ 


3645 


c»rf"«« 


— 


30510 


h^dep 


+ 


120 


b'dp 


— 


270 


h^d^dp 




• • • 


crf»« 


+ 


7290 


dd^P 


— 


2235 


ddep 


— 


3150 


d'dp 


— 


1890 




— 


729 


dddp 


— 


2310 


ddp 


+ 


3420 


d'd*ep 


+ 


2700 


^^\?cP 


— 


5 


ddp 


+ 


10755 


dd^P 


+ 


2920 


d^ddp 


+ 


7590 


deP 


+ 


15 


dd'eP 


+ 


10625 


dd^dp 


— 


18000 


d^df 


+ 


8256 


/^' 


— 


10 


dd^dp 


— 


26700 


dddp 


+ 


43800 


dd'P 


— 


105 


IPc^P 


+ 


10 


dddp 


+ 


795 


ddf 


+ 


5030 


dd'dp 


— 


14360 


ed'P 


— 


120 


ddf 


— 


10070 


dd^eP 


+ 


32280 


dd'df 


— 


43605 


cd^P 


+ 


420 


dd^P 


+ 


180 


dd^dp 


— 


81840 


ddd 


— 


12310 


c^p 


— 


280 


dd^dp 


+ 


1950 


dd^df 


— 


85800 


dd^ep 


+ 


4755 


d^ep 


— 


240 


ddPdp 




• • • 


ddd 


— 


28710 


dd'df 


+ 


77790 


d^^p 


+ 


210 


dd^df 


+ 


36510 


dd^P 


+ 


1260 


dd^d 


+ 


59835 


df?P 




• • • 


ddd 




• • • 


dd^df^ 




« • • 


diC^f^ 




• > • 


Jf" 




• • • 


dd^ep 


+ 


25880 


dd'df 


+ 


181980 


dd^df 


— 


57060 


\^edp 


+ 


200 


dd^dp 


— 


32370 


dd^d 


+ 


153480 


dd^d 


— 


114960 


&^p 


— 


40 


dd^df 


— 


12180 


dd^ep 


— 


26700 


dd^ef 


+ 


19020 


^d^ep 


— 


1140 


ddFd 


— 


9850 


dd^df 


— 


41360 


dd^d 


+ 


109660 


^deP 


— 


480 


cd^P 


+ 


195 


dd^d 


— 


306900 


dd}^f 


— 


2481 


e^p 


+ 


1040 


ed'dp 


— 


43800 


dd^P 


+ 


14360 


dd^d 


— 


56110 


cd^P 


+ 


660 


cd^df 


+ 


72755 


dd^df 


— 


16170 


dd}^t 


+ 


14895 



302 



TABLES OF THE CO VARIANTS M TO W OF THE BINARY QUINTIC. [143 



W, 29 A (continued). 



a» 6 VeP 


_^_ 


1620 


a> b^d^P 


+ 


5575 


a^ h'cd^^ 


+ 


41250 


a> b^<^P 


+ 


5580 I 


a' b'^f 


+ 


1 


d'e'P 


— 


5030 


d^f 


+ 


5445 


c'^d^P 


— 


9540 \ 


b^ce/^ 


+ 


10 


d^t'f 


— 


4255 


d}^^ . 


— 


6525 


c'^d'ey* 





2340 1 


d^P 


+ 


20 


d^^ 


+ 


2175 


}^<?ep 


— 


900 


i^'^d^f 


+ 


20610 1 


d^P 


— 


130 


b^i*dp 


— 


1110 


<?d}p 


— 


510 


c^V 


— 


4350 1 


^p 


+ 


90 


cV/* 


+ 


870 


(»dep 


+ 


120 


<?d^P 


+ 


45130 1 


b^i^ap 


— 


65 


&d^ep 


— 


5550 


^t'P 


+ 


23790 


d'd^d'f 


— 


92200 


c»ey» 


— 


165 


<*df?P 


+ 


24240 


ed^tp 


— 


32000 


&d^d^ 


— 


25050 


cd^eP 


+ 


870 


&^P 


+ 


16194 


f?d^^p 


+ 


58080 


e^P 


— 


19020 


cd^P 


— 


670 


d'd'P 


t- 


1240 


^d^f 


— 


15440 


i^d^^f 


+ 


46050 


Ci^P 


+ 


180 


i^d^^P 


— 


45180 


cV 


— 


12500 


cWj* 


+ 


138750 


d^P 


— 


45 


d'd't^P 


+ 


12180 


<^d^P 


— 


48360 


dd!ef 




• • • 


d^e^P 


+ 


75 


i^d^f 


— 


66650 


i^d'i^P 


+ 


41360 


dd^i? 


— 


178200 


d^e^P 


— 


135 


cV 


— 


8550 


d'd^^f 


— 


181600 


^d?f 


— 


1650 


d^P 




• • • 


&d^eP 


— 


17520 


c»^^ 


— 


18400 


c»cf^«» 


+ 


103950 


ey 




• • • 


<?d'^p 


+ 


91260 


&d^eP 


+ 


180600 


&d}^t 


— 


30250 


b'c'P 


+ 


30 


c'd^e^f 




• • • 


&d^i?f 




• « • 


c*cP 


+ 


3600 


<?deP 


— 


280 


c'd'e' 


+ 


62100 


c»c^e» 


+ 


289800 


b^i^eP 




• • • 


<?i?P 


+ 


2080 


i^d?p 


— 


22600 


c'd^P 


— 


77790 


d^d'P 


+ 


1215 


&d?P 


— 


1320 


i^d^^P 


+ 


85800 


i^i?f 


— 


87000 


d^d^P 


— 


270 


i?d^^P 


— 


3000 


i^d^e'f 


— 


148890 


c*rf««* 


— 


318500 


d^f 


— 


5445 


i^de^P 


+ 


4880 


c'd*^ 


+ 


1850 


<?d^ef 


+ 


92200 


c^'cPeP 


— 


5580 


i^i^P 


— 


4320 


cd^ep 


— 


150 


c'd'^ 


+ 


179500 


c"cW/ 


+ 


17520 


cd^eP 


+ 


2760 


cd^d'f 


+ 


15440 


c'd''/ 


— 


17520 


c"cfo» 


+ 


8700 


cd^^P 


— 


6960 


cd^^ 


+ 


10350 


c'd'^^ 


— 


69000 


d'd^p 


+ 


2481 


cd^^P 


+ 


6480 


^ioy2 


— 


8256 


cd^e 


+ 


15300 


d^d^i?f 


— 


10595 


cd^f 




• • « 


<^^/ 


+ 


12210 


d"* 


— 


1350 


<^d^^ 


— 


31150 


c^ 




• • « 


c/»c^ 


— 


7050 


b^c'^dp 


+ 


135 


dd^tf 


+ 


1650 


d^P 


— 


1390 


b'^P 


+ 


225 


c^^^P 


+ 


540 


i^d^d 


+ 


37950 


d^^P 


— 


600 


c'deP 


+ 


3600 


&d^ep 


+ 


8820 


dd^f 




• • • 


d*^P 


+ 


10070 


c'^P 


— 


8100 


(?d^P 


— 


41640 


ddPt? 


— 


22275 


d^i^f 


— 


12600 


(^d^P 


+ 


940 


i^f^f 


— 


12210 


ddf'e 


+ 


6600 


c^e» 


+ 


4050 


i^d^^P 


+ 


12180 


i»d*P 


+ 


30265 


dd^ 


— 


800 


b^^ep 


— 


30 


d'ilep 


— 


72755 


&d?^P 


+ 


16170 


a« &"«/• 


— 


5 


i^d^P 


+ 


1995 


c«.y 


+ 


4255 


i^d^e'f 


+ 


62025 


b'^edp 


+ 


10 


d'd^P 


— 


1795 


(*d*ep 


+ 


. 46880 


d'dt^ 


+ 


44225 


edp 


+ 


75 


c^eV 


— 


9875 


<^(P^P 


— 


360 


dd^ep 


— 


141240 


d^fP 


— 


130 


i c»c/»c/* 


+ 


3220 


(^d^/f 


+ 


148890 


dd'^f 


+ 


87000 


ddp 


+ 


315 


i?d?^P 


+ 


5240 


&d^ 


+ 


38950 


dd^^ 


— 


129000 


dp 


— 


216 


&d^P 


+ 


4180 


c^'d^P 


+ 


42330 


i^d'P 


+ 


57060 


l^dP 


. — 


5 


\ <^eV 


+ 


12600 


i^d^^P 


• — 


181980 


d'd^^f 




• • • 


ddep 


— 


30 


! c'dJ'P 


+ 


1275 


c'dVf 




• • « 


c»c^«* 


— 


5250 


ddp 


— 


705 


i?d^^P 


+ 


17670 


c*d'd' 


— 


220125 


dd^ef 


— 


46050 


cd^P 


+ 


260 


c'd^ep 


— 


36510 


&d^eP 


— 


63960 


&d^i? 


+ 


122800 


ed^dp 


— . 


265 


&d}^f 




• • • 


&d^^f 


+ 


181600 


e'dy'f 


+ 


10595 


ed^P 


— 


990 


c^d^ 


— 


6075 


ed^^ 


+ 


159000 


c^d^^ 


— 


88125 


c^P 


+ 


1620 


cd^eP 


+ 


1820 


f?d^P 


+ 


43605 


i^d'^e 


+ 


27300 


d^P 


._ 


555 


cd^^p 


— 


38820 


i?d^^f 


— 


62025 


c«cP 


— 


3375 


d^P 


+ 


1620 


cd^^f 


+ 


66650 


c'd'e' 


— 


92500 


b^^P 




• • * 


d'dP 


— 


1215 


cd^e' 


— 


19800 


Cd^'^ef 


— 


20610 


c^^deP 


— 


1080 


ddf 




« • • 



143] 



TABLES OF THE COVAEIANTS M TO W OF THE BINARY QUINTIC. 



303 



W, 29 A (concluded). 



»<^ w 




• • • 


a« 6V(^/» 


+ 


34340 


d^ h^i^d^ef 


— 


138750 


a<» 6 We» - 


17875 


Wtp 


+ 


30 


&d^i?P 


— 


153480 


c^dd' 


— 


1250 


ddf - 


6600 


ed^p 


— 


370 


c»c^ey 


+ 


220125 


i?d}^f 


+ 


31150 


dd'd + 


4125 


edi^f' 


+ 


1800 


i?di^ 




• « • 


d'dJ'd' 


+ 


40000 


dd^e 


• • • 


^t'p 


+ 


2845 


i?d?eP 


+ 


6660 


c'd}^e 


— 


18750 


dd^ 


• • • 


c»d»e/* 


+ 


570 


^d^f 


+ 


18400 


cd^ 


+ 


2250 


h^d^P + 


729 


c»(^ey» 


+ 


1640 


i?d^^ 


— 


73375 


U'c^^P 


— 


135 


d*dep - 


3645 


^d^P 


— 


4185 


cd^P 


+ 


12310 


c'^'dep 


— 


12060 


d^df + 


1350 


i?^f 


— 


4050 


cd^e^f 


— 


44225 


c^V/» 


— 


1980 


d'^dp + 


1620 


cd^P 


+ 


1110 


cd'f^ 


+ 


42500 


&d^P 


— 


69220 


d^ddf + 


3375 


cd^^P 


— 


4170 


d^'ef 


+ 


4350 


d'd'^P 


+ 


89550 


d^dd - 


2250 


cd*t^P 




• • • 


d»e» 


— 


5125 


d'de'f 


— 


41250 


d^def - 


3600 


cd^^f 


+ 


6075 


WeP 


— 


45 


d'd' 


+ 


5125 


d^dd + 


2125 


afo" 




• • • 


ed^p 


— 


2940 


ed^eP 


+ 


240975 


d^df + 


800 


d^eP 


+ 


3620 


ed^p 


— 


9960 


d'd^^f 


— 


179500 


dH^d - 


500 


^^P 


— 


9310 


<?eP 


— 


8175 


d^d^d' 


+ 


80125 


d^de 


• • • 


rf*cy 


+ 


8550 


(^dJ'eP 


— 


46160 


dd^P 


— 


109660 


dd 


• ■ • 


^^ 


— 


3375 


d^d^d'P 


+ 


34300 


dd^f 


— 


122800 






Vf^dp 


+ 


210 


i^d^f 


— 


10350 


c'rfV 


+ 


1250 






ff^p 


— 


615 


i?^ 


+ 


7375 


d'def 


+ 


178200 






f^d^ep 


— 


1285 


&d^P 


— 


73828 


d'd^^ 




• • • 






c'd^P 


— 


11420 


&d>i?P 


+ 


306900 


dd^f 


— 


37950 






c*cy» 


— 


3765 


&de'f 


— 


159000 


dd^d" 


— 


37125 






i^d'P 


— 


3155 


i^d^d' 


+ 


73375 


d'd^^e 


+ 


17875 






&i^i?P 


+ 


3160 


i^d^ep 


+ 


71610 


d'd^ 


— 


2125 






&d^^p 


+ 


9850 


&d^^f 


— 


289800 


¥d^ep 


+ 


1620 






(^d^f 


+ 


19800 


<^d^^ 




• % % 


d'dp 


+ 


30510 






&^ 


+ 


3375 


^d^P 


— 


59835 


d^dd'P 


— 


15120 






ed^tp 


— 


13500 


(^dd^f 


+ 


129000 


d'i^f 


+ 


6525 






ed>^p 


+ 


50550 


C>(^6* 


+ 


80500 


d'^d^ep 


— 


128490 






ed^^f 


— 


62100 


^d^tf 


+ 


25050 


d'^d'e'f 


+ 


69000 






ed"^ 




• • « 


c«^e» 


— 


80125 


d^dd 


— 


19875 






cd^P 


— 


7240 


cd}'f 


— - 


8700 


ddp 


+ 


56110 






c^^P 


+ 


28710 


crf^V 


+ 


19875 


dd^ey 


+ 


88125 






cd^t^f 


— 


38950 


cf'e 


— 


1125 


ddd 


— 


40000 






cd^^ 


+ 


25875 


h^dp 


+ 


990 


dd^ef 


— 


103950 






d^eP 


— 


6155 


i^^P 


+ 


1710 


ddd 


+ 


37125 






(fey 


+ 


12500 


^d'ep 


+ 


34620 


dd^f 


+ 


22275 






d*^ 


— 


7375 


f»d^P 


+ 


4650 


ddd 




« • • 






h^ep 


— 


45 


i?^f 


+ 


7050 


dd?t 


— 


4125 






d^dep 


+ 


615 


ed'P 


+ 


92290 


ddP' 


+ 


500 






c»«y» 


+ 


3880 


ed^^p 


— 


243000 


h'd^dP 


— 


7290 






&d^P 


+ 


4300 


ed^^f 


+ 


92500 


d^dp 


+ 


810 






&d^^P 


+ 


13430 


edd^ 


— 


42500 


d^d'eP 


+ 


34155 






&d^P 


+ 


18750 


<^(^ep 


— 


219730 


d^ddf 


— 


15300 






&^f 


— 


2175 


i*d^^f 


+ 


318500 


d^d 


+ 


1125 






i^d^eP 


+ 


30040 


d'd^ 


— 


80500 


d'd'P 


— 


14895 






^d^e'P 


— 


101450 


&d:^p 


+ 


114960 


d'd^df 


— 


27300 






t^d^^f 


— 


1850 


&^^f 


+ 


5250 


d^d^d 


+ 


18750 






d'dfP 


— 


25875 


c»cfe* 




« • • 


d'^dtf 


+ 


30250 







304 TABLES OF THE COVARIANTS M TO W OF THE BINARY QUINTIC. [l43 

For the lower covariants the numerical verifications are given for the entire 
coeflScient, but for the higher ones where the number of terms in a coeflScient is con- 
siderable they are given separately for the diflferent powers of a; and it is also 
interesting to consider them for the separate combinations of a and b. I recall that the 
positive and negative numerical coefficients are summed separately, so that (+a number) 
means that the sum of the positive numerical coefficients is equal to the sum of the 
negative numerical coefficients and thus that the whole sum is =0. 

It is to be observed that for the lower covariants the sums of the numerical 
coefficients do not vanish for the separate powers of a: thus in the invariant 0, 141, 
the sums of the numerical coefficients for the terms in a\ a^ a® are =1, — 2, 1 
respectively. 

As regards the invariants Q and Q\ for the first of these, Q, the sums of the 
numerical coefficients for the terms in a*, a', a', a\ aP are each of them =0, but this 
is not the case as regards Q' ; in fact Q' is = G' 4- a multiple of Q ; hence the sums 
for Q are the same as those for G*, viz. they are =1, —4, +6, —4, +1 respectively. 
Like results present themselves in other cases, and they might probably be accounted 
for in a similar manner; we have a series of sums not each =0, but which are equal 
to a set of binomial coefficients taken with the signs + and — alternately and thus 
the sum of these sums is = 0. 

For R, 8 and 8\ I have given the sums for the different powers of a; and 
in regard to iS I give here the following paragraphs from the Tenth Memoir on 
Quantics : — 

I remark that I calculated the first two coefficients So, Si, and deduced the other 
two, Sa from 8i, and S, from 8o, by reversing the order of the letters (or which is 
the same thing, interchanging a and /, b and e, .c and d) and reversing also the signs 
of the numerical coefficients. This process for flfj, flf, is to a very great extent a veri- 
fication of the values of So, 8i. For, as presently mentioned, the terms of So form 
subdivisions such that in each subdivision the sum of the numerical coefficients is 
= : in passing by the reversal process to the value of flf,, the terms are distributed 
into an entirely new set of subdivisions, and then in each of these subdivisions the 
sum of the numerical coefficients is found to be = 0; and the like as regards Si and S^. 

If in the expressions for So, S^ 8^, S, we first write d = e=f=l, thus in eflfect 
combining the numerical coefficients for the terms which contain the same powers in 
a, 6, c, we find 

So = a' (- 2c» + 6c» - 6c 4- 2) 

-h a« {6»(6c« - 12c - 6) + 6(- 15c' 4- 33c»- 21c 4- 3) 

4- b' (42c* - 147c» 4- 195c" - 117c 4- 27)} 
4- a {b*. 4- 6» (30c* - 36c 4- 6) 4- 6' (- llTc* 4- 249c« - 183c 4- 51) 

4- 6 (9c« 4- 148c* - 378c' 4- 330c» - 99c) 4- b' (- 63c« 4- 166c* - 147c* 4- 45c»)} 



143] TABLES OF THE CO VARIANTS M TO W OF THE BINARY QUINTIC. 305 

+ a^{6^2 + 6»(-15c + 3) + 6*(76c>-69c + 24) + &»(-9c*-167c'-i-225c«-87c-2) 
+ 6» (72c» + 48c* - 186c» + 96c») + 6 (- 126c« + 201c» - 87c*) 
4- 6* (27c« - 4f6c' + 20c«)} 
which for c = 1 becomes 

= 26« - 126» + 306* - 406» + 306« -126 + 2, that is 2 (6 - 1)«, 

and for 5 = 1, becomes =0. 

8, = a» (Oc» + Oc + 0) 

+ a« {6» (Oc 4- 0) + 6 (3c^ - 9c» 4- 9c - 3) + 6* (24c* - 99c» + 163c» - 105c + 27)) 
+ a {6*.0 + 6'(-6c« + 12c-6) + 6''(-24c« + 90c''-108c + 42) 

+ 6 (33c* - 90c» 4- 54c» 4- 30c - 27) 4- 6* (- 27c« 4- 78c« - 66c* 4- 6c» 4- 9c»)} 
+ a« {6* (3c - 3) 4- 6* (- 1 5c 4- 15) 4- &» (6c» - 1 2c'» 4- 36c - 30) 

4- 6' (9c» - 42c* 4- 84c» - lOBc^ + 57c) 4- 6 (9c« - 54c» 4- 96c* - Sic*) 
4- b' (9c' - 9c«)} 
%vhich for c = l becomes =0. 

/S% = a» (Oc 4- 0) 

4- a» {6» . 4- 6 (Oc« 4- Oc 4- 0) 4- 6' (18c* - 72c^ 4- 108c* - 72c 4- 18)} 
4- a l&»(0c4-0)4-6»(-33c» + 99c«-99c4-33)+6(57c*-162c»4-144c«-30c-9) 
4- 6" (- 60c« 4- 207c* - 261c» 4- 141c" - 27c)} 

4- a« {6» . 4- 6* (15c« - 30c 4- 15) -f 6» (- 54c» 4- 102c« - 42c - 6) 

4- 5» (123c* - 297c» -f 243c« - 87c 4- 18) 4- 6 (- 27c« + 102c* - 96c» 4- 21c=) 
4- 6" (27c' - 60c« 4- 51c« - 12c*)} 

'^'V'hich for c = l becomes =0. 

^= a».0 

4- a* {5 (Oc 4- 0) 4- 6« (Oc» 4- 0c» 4- Oc 4- 0)} 
4- a {6».0 4-5»(0c"4-0c4-0)4-6(-9c*4-36c'-54c*4-36c-9) 
4- &• (36c» - 171c* + 324c» - 306c« 4- 144c - 27)} 

4- a' {6* (Oc 4- 0) 4- &» (7c» - 21c« 4- 21c - 7) 4- 6»(- 39c* 4- 135c'- 171c» + 93c - 18) 
4- b (66c* - 243c* 4- 333c» - 201c» 4- 45c) 
4- 6^ (- 27c' 4- 101c« - 141c« 4- 87c* - 20c»)} 

'^^liich for c = l becomes =0. 

It follows that for c = rf = e=/=l, the value of the co variant S is =2(6 — 1)V, 
"^^liich might be easily verified 

C. n. 39 



306 TABLES OF THE COVARIANTS M TO W OF THE BINABT QUINTIC. [143 



For T, U, V and W, I look at the sums for the different combinations of a and £ 
Thus for T we have 





a; 


coefficient. 






y coefficient. 


a*b^ 






26 


26 


a^b'^ 


-= 


12 


a»6> 




* 


14 


M\/ 


a»6« 


* 


2 


b^ 






141 




6* 




112 


b^ 






281 


436 


6« 




281 


aH* 




^ 


1 


zUV/ 


a»6» 


A 


42 


6» 






106 




6» 




546 


6« 






186 




b' 




696 


b' 






1173 




b' 




366 


b' 






2272 


3738 








a^l^ 




* 


16 


*j t tf\j 


a} I/' 


* 


5 


6< 






359 




b* 




179 


6» 






1411 




b* 




821 


6« 






3103 




6' 




2097 


b^ 






3030 




b" 




2147 


b^ 






1197 


9116 


b' 




1262 


a«6' 




«^ 


2 


«7 X X V 








6« 


92 


— 


78 




an« 


A 


28 


6» 


307 


— 


349 




6» 




342 


6^ 


1073 


— 


1003 




b* 




1790 


6» 


2040 


— 


2110 




6» 




3496 


6« 


1930 


— 


1880 




6« 




3445 


6> 


1207 


— 


1221 




6> 




2064 


6« 


231 


— 


239 


6880 


6<> 




463 

-i 















12 



395 



1650 



6511 



11628 



A 20196 ^ 20196 

Observe here that in the ^^-coefficient for the terms in a® the successive sums 
are -2, 4- 14, -42, +70, -70, +42, -14 + 2, which are the coefficients of -2(tf-iy. 



TABLES OF THE COVARIANTS M TO W OF THE BINARY QUINTIC. 307 



^r U we have 










^ 36 




^ 24 




b' 


198 




b' 


242 




^ 2 




6' 


208 




b' 


286 




b' 


866 




6^ 


1246 




^ 64 




b* 


328 




6» 


1258 




b^ 


2586 




b' 


2186 




b' 


856 




dc 4 




b' 


70 




6» 


448 




b* 


1488 




6» 


2140 




6= 


1678 




b' 


884 




b' 


166 



36 



464 



2608 



7278 



6878 
17264 



39—2 



308 



TABLES OF THE COVABTANTS M TO W OF THE BINARY QUINTIC. 



[143 



For V we have 





X coefficient. 


a»6« 




^ 36 


a*b* 


^ 20 


b' 




284 


b' 




1094 


a'b* 


2 


6» 




184 


6» 




1656 


b' 




3624 


b' 




4898 


an» 


* 14 


6* 




666 


6» 




6608 


6« 




10512 


b^ 




22042 


b^ 




9162 


a^b"" 


4 


6« 


76 


48 


6» 


2956 


- 3040 


6* 


11946 


^ 11806 


6» 


23924 


- 24064 


6« 


25110 


- 25026 


b' 


25524 


- 25552 


6« 


8822 


- 8812 


a«6« 


18 




6^ 


184 


324 


6« 


4098 


- 3622 


6» 


19350 


- 20274 


6* 


42398 


- 41278 


6» 


51872 


- 52740 


6« 


44320 


- 43900 


6^ 


20624 


- 20740 


b^ 


3870 


- 3856 



36 



1398 



10364 



49004 



=t 98358 



db 186734 
345894 



a»6« 
b"" 



y coefficient. 

^ 24 



4 
144 
436 



a»6» 




^ 


24 


6» 






776 


¥ 






2696 


¥ 






1264 


a»6» 


^ 


6 


b* 






300 


6» 






2236 


6« 






8616 


¥ 






15442 


¥ 






33044 


a}b' 


^ 


78 


6» 






852 


b* 






8310 


b^ 






30200 


6» 






56740 


b' 






39956 


b' 






17986 


aPb^ 






2 


V 


286 


— 


270 


6« 


2026 


— 


2082 


6» 


9360 


— 


9248 


¥ 


19760 


— 


19900 


b^ 


36442 


— 


36330 


b* 


30340 


— 


30396 


b' 


23426 


— 


23410 


b' 


5120 


— 


5122 



24 



584 



4760 



59644 



154122 



126760 
345894 



Here in the ^^-coefficient for a^ the successive sums are — 4, +28, — 84, 4- 140,- 
— 140, + 84, — 28, + 4, which are the coefficients of — 4 (tf — 1)' ; and for a® the successive 
sums are 18, -140, +476, -924, +1120, -868, +420, -116, +14, which are the 
coefficients of 18 (tf — 1)^ + 4 (tf — 1)'. In the y-coefficient the successive sums are 
-2, +16, -56, +112, -140, +112, -66, +16, -2, which are the coefficients of 
-2(tf-l)». 



] TABLES OF THE C0VARIANT8 M TO W OF THE BINARY QUINTIC. 



309 



Finally for W we have 



2972759 



7A0 



a' 6 



16 



aH 


-fc 175 


h^ 


806 


a^h" 


* 80 


6« 


1175 


6» 


2760 


¥ 


6871 


a*b* 


Ik 570 


b^ 


5200 


6« 


18005 


6* 


44720 


b' 


23810 


a*6* 


't^ 90 


6» 


2386 


b* 


26675 


6* 


84680 


6« 


107730 


b" 


199160 


b' 


240499 


a«6' 


* 15 


6^ 


640 


&• 


8260 


6» 


59135 


6< 


182055 


6« 


341470 


6» 


699260 


6» 


612015 


b^ 


304501 



16 



981 



10886 



92305 



661220 



a^6»« 


+ 1 




6^ 


120 - 


130 


6» 


1125 - 


1080 


6' 


30350 - 


30470 


6« 


122400 - 


122190 


6» 


332494 - 


332746 


6* 


729150 - 


728940 


6» 


880750 - 


880870 


6« 


466935 - 


466890 


6^ 


363670 - 


363680 


60 


76116 - 


76115 


a«6" 


+ - 


5 


b'^ 


400 - 


346 


6» 


3500 - 


3765 


6» 


26240 - 


25460 


6^ 


154030 - 


155560 


6« 


409700 - 


407600 


6» 


747985 - 


750043 


6* 


745920 - 


744480 


6» 


613100 - 


613805 


6» 


311790 - 


311560 


6^ 


89215 - 


89260 


bP 


9999 - 


9995 



3003111 



3111879 
9087749 



2207351 



Here for the terms in a* the successive sums are 

1^ _10, +46, -120, +210, -252, +210, -120, +45, -10, +1, 

;h are the coefficients of (tf- ly^ and for the terms in a* the successive sums are 
«5, +54, -.265, +780, -1530, +2100, -2058, +1440, -705, +230, -45, +4, 
± are the coefficients of - 5 (tf - 1)" - (« - ly. 



310 



[144 



144. 



A THIRD MEMOIR UPON QUANTICS. 



[From the PhiloaophiccU Tramactiona of the Royal Society of London, voL cxlvi. for the 
year 1866, pp. 627—647. Received March 13,— Read April 10, 1856.] 



Mt object in the present memoir is chiefly to collect together and put upon 
record various results useful in the theories of the particular quantics to which they 
relate. The tables at the commencement relate to binary quantics, and are a direct 
sequel to the tables in my Second Memoir upon Quantics, voL CXLVI. (1856), [141]. 
The definitions and explanations in the next part of the present memoir are given 
here for the sake of convenience, the further development of the subjects to which 
they relate being reserved for another occasion. The remainder of the memoir consists 
of tables and explanations relating to ternary quadrics and cubica 

Covariant and other Tables, Nos. 27 to 50 (Nos. 1 to 50 binary quantics)^ 

Nos. 27 to 29 are a continuation of the tables relating to the quintic 

(a, 6, c, d, e, fjx, yf. 

No. 27 gives the values of the different determinants of the matrix 

( a, 46, 6c, 4d, e ) 

a, 46, 6c, 4d, e 

6, 4c, 6d, 4«, / 
6, 4c, 6d, 4«, / 
determinants which are represented by 1234, 1235, &c., where the numbers refer to 

1 The Tables 49 and 60 were inserted October 6, 1S56.— A. C. 



144] 



A THIBD MEMOIR UPON QUANTICS. 



311 



the different columns of the matrix. No. 28 gives the values of certain linear 
functions of these determinants, viz. 



2,= 


1256 + 


2345- 


- 2 . 1346, 


L'^i 


. 1256 - 


1346, 




8M = - 


• 1345 + 2 


.1246, 




8ir=- 


2346 + 2 . 


, 1356, 




8iV=- 


' 1245 + s 


. 1236, 




8N' = - 


2356 + 8 . 


. 1456, 




80P = 


L'-3L 


= 8. 


1346 - 8 


16P' = - 


5L'- L 


= — 18. 


1256 - 8 



^t the end of the two tables there are given certain relations which exist between 
tr^he terms of Tables 14, 16, 25, 26, 27 and 28. 



No. 27. 



1234. 


12d6. 


1236. 


1246. 


1246. 


1345. 


1256. 


2345. 


«V 


aV- ^ 


a*d/+ 6 


a'd/- 6 


a V + 4 


a'c/" 


ay«+ 1 


ay« 


«»« - 16 


a^de + 24 


aV 


aV + 16 


oi^- 4 


a6^- 24 


o^- 2 


a6«/* 


«V + 36 


aiy+ 4 


abc/^ 22 


abr/+ 6 


a6e«- 4 


abe^ + 64 


occf/"- 16 


acd/+ 20 i 


^3i«e+ 16 


abee- 84 


abde- 6 


ahde- 26 


o^y- 24 


acy+ 24 


octf« + 16 


ace* - 80 ; 


^3bcd- 152 


oW"- 24 


ac^e + 16 


ac«« - 96 


acde-\- 24 


acde- 208 


acPe + 16 


ad^e+ 60 


«j* + 96 


a(j«rf+ 64 


acd^ 


acd^+ 96 


ocP 


ew^ + 144 


6V/- 15 


6*^- 80 


*(i + 80 


b^e + 60 


by + 16 


6y ... 


6»c/ + 24 


6V 


6»62 ... 


6V + 240 ; 


«V - 60 


i^cd- 40 


b'ce - 10 


b^ce + 90 


bHe - 20 


6»c^- 40 


6cy ... 


be"/ + 60 ; 




OCT ... 


6«rf« ... 


6»c?» - 80 


oc^c ... 


6c»6 + 60 


6crfc 


6cde- 860 1 






bc^d 


bc^d 


6ccP 


bed"- 40 


6rf» 


6c^ + 960 ■ 


1 




C/ • . . 


(j . • • 


c»c/ 


c'c/ 


c»« 


c»e + 960 j 


i 

1 












c'd' ... 


c'd' - 320 j 



1346. 


2346. 


1356. 


2356. 


1456. 


2456. 


3456. 


ay* ... 


ai^ 


a^+ 4 


acP- 6 


axp + 6 


a<^«- 4 


a^" 


a6c/'+ 16 


ac«/*- 24 


cLcef — 4 


adtf-v 6 


adef- 22 


acy+ 4 


6(^- 16 


acc(/'- 36 


a(^/+ 24 


a(/y- 24 ae* 


a<j» + 16 


6c/« + 24 


6«y + 16 


ac«« - 16 


a{2s* 


a^ + 24 ^»y« + 16 


6y* ... 


bdef-^ 84 


cy« + 36 


ad^e + 36 


6««/+ 64 


V'tf- 4 


6cc/- 26 


bcef^ 6 


ftc» + 60 


cdef- 152 


l^df-- 16 


bcdf-^ 208 


6<j(^+ 24 


6«((/-- 96 


6c^/+ 16 


c»c/ - 24 


ce» + 80 


6V ... 


6<j«« - 40 


6ce« - 20 


6de» + 90 


6<^- 10 


ccZy+ 64 


c^/ + 96 


6cy+ 36 


6rf«e + 60 


bd^e 


i^df + 96 


d'df ... 


cc^ - 40 


d?^ - 60 ! 


bcde- 20 


c»/ + 144 


cy ... 


e^ - 80 


vTVi ... 


d^t 


1 


W» 


c»cfe- 40 


c*cfe 


ccPtf 


cc^e 






V c ... 


CflP 


CflP 


cf* 


d^ 






<?d^ 






1 







312 



A THIRD MEMOIR UPON QUANTICS. 



[144 



No. 28. 



N. 


M. 


L. 


L'. 


P. 


F. 


M'. 


N'. 


aHf + 3 
a V - 2 
ahcf- 9 
abde+ 1 


a V + 1 
abd/+ 2 
a6c»- 9 
ocy- 9 


ay « + 1 

a6c/- 34 
aor//*+ 76 
acc« - 32 


ay«+ 3 
oic/- 22 
acdf- 12 
occ' + 64 


ay> ... 

ahef+ 1 
oo//- 3 
oc^ + 2 


ay«- 1 

aA«/'+ 9 
occ^- 1 
aci^ ^ 18 


abf-^ 1 
ac«/*+ 2 
(w^/- 9 
a<i«»+ 6 


ac/^+ 3 ^ 
oic/- 9 

oc* + 6 \ 

6y«- 2 


cu^e + 18 


acde-i- 32 


ad'e- 12 


a<Pe- 36 


cuPe 


o(f"e+ 12 


bhf- 9 


bee/ + 1 


acd^- 12 
b*/ + 6 


o^ - 18 
6 V + 6 


6V/- 32 
6V + 225 


6V/+ 64 
6V - 45 


b'd/+ 2 
6V - 9 


b^df- 18 
6V 


6«i/'+ 32 


6dy+ 181 
W«» - 1^ ; 


b^ce - 15 
b^d" + 10 


b^de 

bc'e - 15 


bc^f- 12 
6cfl?c- 820 


bey- 36 
6ccfe+ 20 


bc\f ... 
6<;flfe+ 31 


6cy + 12 
ftcSc + 45 


bd'e- 15 
cy - 18 


c^- 12^ \ 
C*(5« + lO^ ! 


6c'cZ 


bcd^-^ 10 


b<P + 480 


M» 


bd* - 18 


6ii» - 30 


c'de + 10 


cd^e ... * 


c* 


C^(/ 


c^e + 480 


CJ C/ • • • 


c«e - 18 


c'e - 30 


ci* 


c^ ... -' 






c'd' - 320 


c"flP 


c'd' + 12 


c«cZ« + 20 







If the coeflBcients of the table 14 are represented by ^A, B, ^C, viz. ¥rritiDg 

il = 2 (oc - 46d + 3c»), 
B = a/- 36e + 2cd, 
(7 = 2(6/-4ce+3d«), 

then we have the following relations between 1234, &c. and A, B, C, viz. 



1 


+ Bx 


+ Ax 


1234 = 


+ 6a« 




-12a6 


+ 16 oc -10 6» 


1235 = 


+ 6 ah 




- 2 oc - 10 6> 


+ 6ad 


1236 = 


- 2ac+ 8 6= 




+ 6<k£- 18 6c 


- 2d/+ 8c» 


1245 = 


+ 18ac 




- 6 CM? - 30 6c 


+ Sae +10 bd 


1246 = 


+ 12 6c 




+ 4a<5- 4 6rf-24c« 


+ 46c + Scd 


1345 = 


+ 24 ac£ 




- 8 oc - 40 6rf 


+ 4 a/ + 20 6c 


1256 = 


- I ae + 4 6rf + 


3c» 


+ I a/ + 5 be - IS cd 


- lb/ + 4cc+ 3dr 


2345 = 


+ 20 oc + 40 6c/ - 


30 c« 


- SO be + 20 cd 


+ 20 6/ +40 cc- 30 (? 


1346 = 


+ 4 06 + S bd + 


6c» 


-36 erf 


+ 4 6/+ 8cc+ 6d^ 


2346 = 


+ 4 a/ + 20 6c 




- 8 6/- 4cc 


+ 24c/ 


1356 = 


+ 4 6c + 8 cc? 




+ 4 6/- 4c«-24rf« 


+ 12cfe 


' 2356 = 


+ 8 6/ + 10 cc 




- 6c/-30cfo 


+ lSd/ 


1456 = 


+ 6cc 




+ 6 c/ -ISde 


- 2c(^+ 8c» 


2456 = 


+ 6 c/ 




- 2((/*-10c» 


+ 6 c/ 


3456 = 


+ 16 df- 10 e^ 




-12 c/ 


+ 6/« 




and the following relations between L, L\ &c. and A, B, C, viz. 





Cx 


+ Bx 


+ Ax 


^= 


- 3 oc + 3 6« 


+ 3 orf - 3 6c 


- lac+ 1 6rf 


3/ = 


- Sad+ 3 6c 


+ 3ac - 3c' 


- l€/+ led 


Z = 


+ 11 oc + 28 6rf-39c» 


+ 1 a/ - 75 6c + 74 cd 


+ 11 6/+ 28 cc- 39 rf« 


Z'- 


- 7ac+ 4 6rf+ 3c' 


+ 3 a/ + 15 6c - 18 erf 


- 76/+ 4cc+ 3rf» 


2iP = 


- lac- 2bd+ 3c» 


+ 3 6c - 3 erf 


+ 1 h/+ 2 ec- Srf" 


7^ = 


+ 3ac- 6bd+ 3c« 


- 1 a/+ 1 erf 


+ 3 6/- 6 oc + 3 rf» J 


if'= 


- I a/ + I cd 


+ 3 6/- 3rf« 


- Sc/+ 3rfc I 


jr= 


- 16/+ 1 cc 


+ 3c/ - 3rfc 


- 3(i^+ 3c» 1 




_ iiiJiMOIIl UPON QUANTICS. 



313 



-» e nave also the following relations between i, L\ &c. and a, 6, c, d, c, /, viz. 

aP -6Jf 4- ci^T =0, 



aN'+2bM''-cr 



+ 3eN 



= 0, 
= 0, 



3cJ\r'-2dif'4- eF+fM=0, 
The quartin variant No. 19 [G] is equal to 

i.e. it is in fact equal to —4 into the discriminant of the quintic No. 14, [A]. 

The octin variant No. 25 [Q] is expressible in terms of the coefficients of Nos. 14 
and 16, viz. A, B, C, as before, and Ja, /8, 7, JS the coefficients of No. 16, [D], i.e. 

a = 3 (ace -ad^- h^e +26c(Z- c»), 
/8= acf-ade- l^f+ bd^ + bee - d^d, 
y = adf—ae^ — bcf+ bde + d^e —cd^, 
S = 3 ibdf" 6e» 4- 2cde - c*/ - d»), 



then No. 25 is equal to 



A, B, C 
a, ^, 7 

The value of the discriminant No. 26, [Q'], is 

(No. 19)«-128 No. 25. [that is Q' = G« - 128Q.] 

We have also an expression for the discriminant in terms of Z, L\ &c., viz. three 
times the discriminant No. 26 is equal to 

[or say 3Q' =] LU 4- UMM' - &4^NN\ 

remarkable formula, the discovery of which is due to Mr Salmon. 

It may be noticed, that in the particular case in which the quintic has two square 
^tors, if we write 



(a, 6, c, d, e, /$a?, y/ = 5 {(p, 5, rjx, y)«}« . (\, /A$a?, y). 



C. II. 



40 



314 



A THIBD MEMOIR UPON QUANTIC8. 



[144 



then 

a = 5\p^, h = 4tpqK 4- J5 V, 

and these values give 

P =Z(63»-pr), 
M = K, lOpq, 

where the value of iT is 



c = (23" +pr)\ + 2pqfi, 
d = 25r\ + {2f +pr)fi\ 



8 (pfi^ - 2qfi\ 4- rVy (/)r - j«)*. 

The table No. 29 is the invariant of the twelfth degree of the quintic, given i^ 
its simplest form, Le. in a form not containing any power higher than the fourth o^ 
the leading coefficient a: this invariant was first calculate by M. Faa de Bruno. 

No. 29. [See U. No. 29, p. 294] 

The tables Nos. 30 to 35 relate to a sextic. No. 30 is the sextic itself ^ 
No. 31 the quadnn variant ; Nos. 32 and 33 the quadricovariauts (the latter of them, 
the Hessian); No. 34 is the quartinvariant or catalecticant ; and No. 35 is the? 
sextinvariant in its best form, i.e. a form not containing any power higher than the- 
second of the leading coefficient a. 

No. 30. 



a^\ 


6 + 6 


c + 15 


d+20 


e + 15 


/+6 



g+l Wx, y)* 



No. 31. 



No. 32. 



ag 


+ 


1 


¥ 


— 


6 


ce 


+ 


15 


d^ 


— 


10 



^16 



a« + 1 
W - 4 
c« + 3 


a/ + 2 
6tf - 6 
cd + 4 


0^+1 
C6 - 9 

(^ + 8 


hg -^ 2 
cf - Q 
de + 4: 


eg + I 
^/- 4 
e» + 3 



:fe4 



±G 



No. 33. 



*9 



^6 



5«, 2/y 



ac + I 


ad+i 


otf + 6 


a/+ 4 


ag+ 1 


6^+4 


eg + ^ 


dg + 4: 


eg + 1 


("-' 


6c -4 


bd+ 4 


6<»+16 


6/+ 14 


C/ + 16 


d/+ 4 


e/-4. 


r-l 




c« -10 


cd- 20 


c« + 5 


cfo - 20 i c« - 10 






1 








<^-20 




i 

i i 



$*. y)' 



^1 



^4 



:l=10 



db20 



db20 



Jk20 



A 10 



^4 



:fcl 



144] 



A THIRD MEMOIR UPON QUANTICS. 



315 



No. Si. 



No. 35. 



aeeg + 1 
acP -1 
ad^g - 1 
adef + 2 
o^ -1 
h'eg -1 

h'P +1 
bcdg + 2 
6ct/^ -2 
6dy-2 
6d;e» +2 






-1 

+ 2 
+ 1 
-3 
+ 1 



12 



a^<P(^ + 1 
a^defg— 6 
aV/» + 4 
aV<7 + 4 
aV/> - 3 
abcd^— 6 
a^«/gr+ 18 
a^/» -12 
aMy^+ 12 
ahde^g-l^ 
aMf + 6 
ao*5^ + 4 
ac»c«^ - 24 
o^dfg-\% 
a^eP + 30 
ac<Peg + 54 
acdy^-12 



acrfey-42 
ace^ +12 
orf*^ -20 
««/'«/• +24 
acP^ - 8 
W^ + 4 

Wg -12 
6y» + 8 

6V^ - 3 
ftW^r +30 
6»c«/> - 24 
b'iPeg - 12 
6«rfy* - 24 
6»dey + 60 
6V - 27 
6cy^ + 6 
hi^deg - 42 



bccPg 

bccPef 

bcd^ 

bdy 

d^eg 
c'def 

ed^e 



+ 60 
-30 
+ 24 
-84 
+ 66 
+ 24 
-24 
+ 12 
-27 

- 8 
+ 66 

- 8 
-24 
-39 
+ 36 

- 8 



:A:665 



The seztmvariant may be thus represented by means of a determinant of the 
dxth order and of the quadrinvariant and quartinvariant. 



5xNo. 35 = 



4-4(asr-66/+15ce- 





a, 26, 3c, 


4d, 


e 


6, 2c, 3d, 


4c, 


f 


c, 2d, 36, 


4/ 


9 


a, 46, 3c, 2d, 


c 




6, 4c, 3d, 2c, 


/ 




c, 4d, 3c, 2/, 


9 




10*) 


a, 6, c. 


d 






6, c, d, 


e 






c, d, c, 


f 






d, c, /, 


9 





The tables Nos. 36 and 37 relate to a septimia No. 36 is the septimic itself; 
^^0. 37 the quartinvariant. 

No. 86. 



( o+l 


6 + 7 


c + 21 


rf+35 


« + 35 


/+21 


9 + f 


h+1 



40—2 



316 



A THIRD MEMOIB UPON QUAMTICS. 



[U4 



No. 37. 



aW - 1 


6rf«A 


- 40 


abgh ' + 14 


bdeg 


- 50 


acjh - 18 


bdp 


- 360 


acg* - 24 


bf?f 


+ 240 


ad^h + 10 


<?eg 


- 360 


ad/g + 60 


c>P 


- 81 


ae^g - 40 


cd^g 


+ 240 


b^h - 24 


cdef 


+ 990 


6y - 25 


ce" 


- 600 


bcfg + 234 


ciy 


- 600 


bceh + 60 


(£«c» 


+ 375 



±2223 

The tables Nos. 38 to 45 relate to the octavic. No. 38 is the octavic itse 
No. 39 the quadrin variant ; Nos. 40, 41 and 42 are the quadricovariants, the last 
them being the Hessian; No. 43 is the cubinvariant ; No. 44 the quartinvariaut, ai 
No. 45 the quintinvariant, which is also the catalecticant. 

No. 38. 



( 



a+ 1 



6 + 8 



c + 28 



(f + 56 



« + 70 



/+56 



^ + 28 



A + 8 



» + l W^^yf 



No. 39. 



No. 40. 



ai 


+ 


1 


bh 


— 


8 


eg 


+ 


28 


df 


— 


56 


c« 


+ 


35 



±64 



ag + \ 


oA + 2 


ai + 1 


bi + 2 


ci + 1 


6/- 6 


bg - 10 


bh - 2 


ch - 10 


dh- Q 


( cc + 15 


cf + 18 


eg - S 


dg + IS 


eg + lb 


d« - 10 


de - 10 


d/ + 34: 


e/ - 10 /« - 10 






e« - 25 







±16 



±20 

No. 41. 



±35 



±20 



±16 



5^, y)^ 



oe + 1 


a/+ 4 


ag + Q 


ah + i 


ax -^^ 1 6t + 4 


ci + 6 


c^i + 4 


et + 1 


6c£ - 4 


be -12 


bf - S 


bg + S 


bh + 12 cA + 8 


c^A- 8 


eA-12 


/A- 4 


c« + 3 


cd + S. 


ce -22 


cf -48 


eg - 22 ety - 48 


c^ - 22 


fg^ 8 


/ + 3 






cf» + 24 


cfe + 36 


df - 36 «/ + 36 


/' + 24 














6^+45 











±4 



±12 



±30 



±48 



±58 



±48 



±30 



±12 



$a-, yf 



No. 42. 



ac + 1 


ad-^r 6 


ac+ 15 


a/'+20 


a^+ 15 


aA+ 6 


1 
ai + 1 6i + 6 ci + 15 


rfi+20 


«» + 15 


/i + 6'j,i + 


6'-l 


6c -6 


6c£+ 6 


6c +50 


6/+ 90 


bg^ 78 


6A+ 34 


cA+ 78 


dh^ 90 


cA + 50 


/A+ 6 


yA-6 A»- 


/ 




c« -21 


ccf-.70 


ci« -105 


c/ + 126 


c^ + 154 


cfy+126 />-105 


/^-70 


^•-21 




( 

1 

1 








(fo-210 


rf/- 14 
6» -175 


e/ - 210 









±1 



±6 



±21 



±70 



±105 



±210 



±189 



±210 



105 



±70 



±21 



±6 



14] 



A THIRD MEMOIR UPON QUANTICS. 



317 



No. 43. 



No. 44. 



aei + I 
a/h- 4 
a^ + 3 
bdi - 4: 
heh + 12 
hfy - 8 
c»i + 3 
cdh - % 
ceg - 22 
cP 4- 24 
J^g + 24 
(ii'/ - 36 
e* +15 



±82 



ac<7t - 1 


hcgh + 3 


cd^i - 2 


acA« + 1 


hdei + 1 


cdeh - 23 


adfi + 3 


6c(A - 10 


c^//*y + 27 


ac&/A - 3 


hag"" + 9 


ce»y + 19 


oeH - 2 


6eVi + 11 


cef - 21 


a€/7t + 1 


hefg - 23 


(^A + 12 


alf + 3 


hp + 12 


(r€g - 21 


oPg - 2 


c'et + 3 


c^/' - 13 


6Vi + 1 


<?fh + 9 


de"/ + 32 


6^/i* - 1 


c«/ - 12 


e* - 10 


6r/i - 3 


1 





±147 



No. 45. 





1 

i 


acegi 


+ 


1 


«/* 


+ 


1 


bde(f^ 


4 


CiPg^ 


+ 1 




t 


aceJi^ 


— 


1 I 


b^egi 




1 


hiiPg 


+ 2 


cdefg 


- 2 






acfi 


— 


1 


IPeh' 


+ 


1 


be>h 


2 


cdp 


2 






acfgh 


+ 


2 


I'M 


— 


2 


beVg 


V 4 


c^g 


3 






acg^ 


-- 


1 ! 


byi^ 


+ 


1 


hop 


- 2 


ce^dh 


^ 4 






(vPql 


— 


1 


by 


+ 


1 


egi 


- 1 


ccT 


^ 3 




1 


(uPh^ 


+ 


1 


bcdgi 


+ 


2 


eh^ 


+ 1 


dH 


t- 1 




1 
1 


adefi 


+ 


2 


bcdli" 


— 


2 


<?dfi 


+ 2 


d'eh 


2 






Ofiegh 


— 


2 


bcefi 


— 


2 


<?d<jh 


2 


<^fy 


2 






a<//Vi 


— 


2 1 


bceffh 


+ 


2 


eeH 


+ 1 


d'e'g 


h 3 




1 


ad/g' 


-u 


2 1 


br/Vi 


+ 


2 


e'efh 


- 4 


it'^P 


+ 3 




1 


a^i 


— 


1 


kf</' 


— 


2 


oV 


+ 2 


: d^f 


- 4 




1 
1 


ae^jh 


T 


o 1 


bdifi 


— 


2 


cyv 


1- 1 


^ 


- 1 






a^g^ 


+ 


1 ' 


bd'gh 


+ 


2 


cd^ei 


- 3 








1 


aepg 





3 1 


bd(^i 


+ 





cd^fh 


+ 2 


















±56 


If we 


write 























i' ■ 

d 

6-12\, 



No. 39 = /, 




No. 43== J, 




No. 44 = ir, 




No. 45 = £, 




lambdaic, viz. 




h . c , d , 


e-12X ' 


c , d , C + 3X, 


/ 


d , e-2\, f 


9 


e+3\, / . g 


I 


f ,9 , h 


• 



\ 



318 



is equal to 



A THIRD MEMOIR UPON QUANTICS. 



[144 



L + 2\K + 3X»J+ 18V/ - 2592V. 



Nos. 46 to 48 relate to the nonic. No. 46 is the nonic itself; Nos. 47 and 48 
are the two quartinvariants, each of them in its best form, viz. No. 48 does not 
contain a', and No. 47 does not contain aci^, the leading term of No. 48. The 
nonic is the lowest quantic with two quartinvariants. 



No. 46. 



a+l 


6 + 9 


c + 36 


c^ + 84 


e + 126 


/+126 


^ + 84 


/i + 36 


1+9 


i+1 




No. 


47. 




No. 48. 




a'/ - 1 


b/^h 




a«/ ... 


bph + 70 




abii + 18 


b/g^ - 720 




dbij 


bfg^ - 45 




act* 


cYj + 432 




acv" + 2 


c^^gf + 27 




achj - 72 


<?gi - 1728 




achj — 2 


i^gi - 52 


\ adgj + 168 


c W 




adgj + 7 


<^h^ + 25 


i adhi 


cdej - 720 




orf/a- 7 


ccfe; — 45 




a^fj - 108 


cdfi + 2160 




a«^ - 5 


cdf% + 23 




aegi - 576 


cdgh+ 4608 




ocgri -22 


ce^A + 22 


aeh^ + 432 


cfh, 




aefi^ +27 


c(H + 70 




afH + 540 


cefh - 2592 




a/«t +25 


cefh -127 




afgh- 720 


eeg" - 5760 




afgh -45 


c«^ + 32 




a^ + 320 


c/V + 4320 




a^ +20 


cpg + 25 




6«A; 


d?j + 320 




6*/*/ + 2 


cO' + 20 




W - 81 


d^'ei - 720 




W - 2 


d^ei - 45 




hcgj 


d'fh- 5760 




6c5|[/ - 7 


cP/A+ 32 




hchi + 648 


d«5^ - 1536 




tcAt + 7 


d'g^ + 47 




hdjj - 576 


dc/^ + 14688 




bdjj - 22 


<*!/& + 85 




bdgi + 792 


de^h + 4320 




W^t + 74 


d^h + 25 




bdh^ - 1728 


dp - 8640 




bd^i^ - 52 


df* - 50 




begh + 2160 


e»<7 - 8640 




ft^^r/* + 23 


e*^ - 50 




b^j + 540 


cy« + 5184 




be'j +25 


ey« + 30 




be/t - 972 






6(2/t -73 





3[a^ y)*- 



:k 41650 



:A=698 



Nos 49, [49 a] and 50 relate to the dodecadic. No. 49 is the dodecadic itself: 
[No. 49 A, inserted in this place, but originally printed in the Fifth Memoir on Quantics, 
is the dodecadic quadricovariant]. No. 50 is the cubinvariant. [The numerical coefficients 
in this last table as originally printed in the Third Memoir were altogether erroneous, 
and the table as here printed is in fact the table No. 60 j^, of the Fifth Memoir on 
Quantics.] 

No. 49. 



a+l 


6 + 12 


c + 66 


rf+220 


c + 495 


/+792 


^ + 924 


A + 792 


i+495 


J +220 


A;+66 


f+12 


m + l 



5«,yr 



144] 



A THIRD MEMOIR UPON QU ANTICS. 



319 



16 



No. 49 A. 



ag^ 1 


ah+ 6 


at + 15 


a;+ 20 


ak-i- 15 


6/- 6 


6<7-30 


bh- 54 


bi- 30 


6; + 30 


c« + 15 


C/ + 54 


eg + 24 


cA - 150 


ci -270 


( d'-lO 


cfo-30 


rf/+150 


dg + 430 


^+270 






e« - 135 


./ - 270 


c^ +495 
/2 - 540 



aZ + 6 
bk+ 54 
c; - 150 
di- 270 
«A + 1080 
fg- 720 



am + 1 

bl + 30 

cA; + 24 

dj -430 

c» +495 

//i +720 

^ -840 



=1:60 



:l=189 



:t460 



rtSlO 



:tll40 



±1270 



brn + 6 
c/ + 54 
dk - 150 
ei - 270 
/» +1080 
gh - 720 



cm + 15 
dl + 30 
cifc -270 
^ +270 
^i +495 
h^ -540 



c/m + 20 
el - 30 
/>fc -150 
<d +430 
Xi -270 



cm + 15 

/I - 54 

i^ifc + 24 

hj +150 

-135 



t« 



/m + 6 
^/ -30 
A>fe +54 
y -30 



gm + I \ 
A/ - 6 ' 

tA; + 15 ty- \n 

j2 ^10 i$*»y)" 



±1140 



±810 



±450 



±189 



±60 



±16 



No. 50. 



agrn + 1 


c/l - 54 


dhi + 270 


oA/ - 6 


c^A; + 24 


e^A; -135 


aik + 15 


chj +150 


^Jj +270 


a/ - 10 


ci« - 135 


c^i + 495 


bfm — 6 


<Pm - 10 


eh* -540 


6^; + 30 


rfe^ + 30 


/S - 540 


bhk -54 


dfk +150 


fgh + 720 


6v + 30 


^^- _430 


^ -280 


cem + 15 







±2200 



Resuming now the general subject, — 



54. The simplest covariant of a system of quantics of the form 

(where the number of quantics is equal to the number of the facients of each 

quantic) is the functional determinant or Jacobian, viz. the determinant formed with 

the differential coeflScients or derived functions of the quantics with respect to the 
several facients. 

65. In the particular case in which the quantics are the differential coefficient8 or 
derived functions of a single quantic, we have a corresponding covariant of the single 
quantic, which covsuriant is termed the Hessian ; in other words, the Hessian is the 
determinant formed vrilth the second differential coeflScients or derived functions of the 
quantic with respect to the several facients. 

66. The expression, an adjoint linear form, is used to denote a linear function 
fe + i7y+ .... or in the notation of quantics (f, i;,...$a?, y,...), having the same facients as 



320 A THIRD MEMOIR UPON QUANTICS. [144 

the quantic or quantics to which it belongs, and with indeterminate coefficients 
(f, i;,...)- The invariants of a quantic or quantics, and of an adjoint linear form, may 
be considered as quantics having (f, 17,...) for facients, and of which the coefficients 
are of course functions of the coefficients of the given quantic or quantics. An invariant 
of the class in question is termed a contravariant of the quantic or quantics. The 
idea of a contravariant is due to Mr Sylvester. 

In the theory of binary quantics, it is hardly necessary to consider the eontra- 
variants; for any contravariant is at once turned into an invariant by writing (y, — d?) 
for (f, 17). 

57. If we imagine, as before, a system of quantics of the form 

(•$^, y, ...)'«, 

where the number of quantics is equal to the number of the facients in each quantic, 
the function of the coefficients, which, equated to zero, expresses the result of the 
elimination of the facients from the equations obtained by putting each of the quantics 
equal to zero, is said to be the Resultant of the system of quantics. The resultant 
is an invariant of the system of quantics. 

And in the particular case in which the quantics are the differential coefficients, 
or derived functions of a single quantic with respect to the several facients, the 
resultant in question is termed the Discriminant of the single quantic; the discriminant 
is of course an invariant of the single quantic. 

58. Imagine two quantics, and form the equations which express that the differen- 
tial coefficients, or derived functions of the one quantic with respect to the several 
facients, are proportional to those of the other quantic. Join to these the equations 
obtained by equating each of the quantics to zero; we have a system of equations, 
one of which is contained in the others, and from which therefore the facients may 
be eliminated. The function which, equated to zero, expresses the result of the 
elimination is an invariant which (from its geometrical signification) might be termed 
the Tactinvariant of the two quantics, but I do not at present propose to consider 
this invariant except in the particular case where the system consists of a given 
quantic and of an adjoint linear form. In this case the tactinvariant is a contravariant 
of the given quantic, viz. the contravariant termed the Reciprocant. 

59. Consider now a quantic 

(•Ja:, y,...)*", 

and let the facients x, y, ... be replaced by Xx + fiX, Xy-h/^F, ... the resulting function 
may, it is clear, be considered as a quantic with the facients (\ fi) and of the form 



144] A THIRD MEMOIR UPON QUANTICfS. 321 

The coefficients of this quantic are termed Emanants, viz., excluding the first coefficient, 
which is the quantic itself (but which might be termed the 0-th emanant), the 
other coefficients are the first, second, and last or ultimate emanants. The ultimate 
emanant is, it is clear, nothing else than the quantic itself, with (X, F, ...) instead of 
(x, y, ...) for facients : the penultimate emanant is, in like manner, obtained from the 
first emanant by interchanging (ar, y, ...) with (X, F, ...), and similarly for the other 
emanants. The facients (X, F, ...) may be termed the facients of emanation, or simply 
the new founents. The theory of emanation might be presented in a more general 
form by employing two or more sets of emanating fii<;ients; we might, for example, 
write \x-\-fi,X'{-vX\ \y •{■ fiY + vY\ . , , for x, y, ..., but it is not necessary to dwell 
upon this at present. 

The invariants, in respect to the new facients, of any emanant or emanants of a 
quantic (i.e. the invariants of the emanant or emanants, considered as a function or 
functions of the new facients), are, it is easy to see, covariants of the original quantic, 
and it is in many cases convenient to define a covariant in this manner; thus the 
Hessian is the discriminant of the second or quadric emanant of the quantic. 

60. If we consider a quantic 

(a, 6,...$a?, y,...)'^, 
and an adjoint linear form, the operative quantic 

(which is, so to speak, a coutravariant operator) is termed the Evector. The proper- 
ties of the evector have been considered in the introductory memoir, and it has been 
in effect shown that the evector operating upon an invariant, or more generally upon 
a contravariant, gives rise to a coutravariant. Any such coutravariant, or rather such 
contravariant considered as so generated, is termed an Evectant In the case of a 

binary quantic, 

(a , b ,...$ar, y)"», 
the covariant operator 

{da, 96,...$y,-^)"* 

may, if not with perfect accuracy, yet without risk of ambiguity, be termed the Evector, 
and a covariant obtained by operating with it upon an invariant or covariant, or 
rather such covariant considered as so generated, may in like manner be termed an 
Mvectant, 

61. Imagine two or more quantics of the same order, 

(a, 6, ...][a?, yf, 
(a, /8,...$a?, yf, 

-we may have covariants such that for the coefficients of each pair of quantics the 
covariant is reduced to zero by the operators 

ad^ + Idp 4 . . . , 

ad a + I3di, -h . . . . 
C. II. 41 



322 A THIRD MEMOIR UPON QUANTICS. [l44 

Such covariants are called CtmbinanUy and they possess the property of being inva- 
riantive, quoad the system, i. e. the covariant remains unaltered to a factor priSy when 
each quantic is replaced by a linear function of all the quantics. This extremely 
important theory is due to Mr Sylvester. 

Proceeding now to the theory of ternary quadrics and cubics, — 

First for a ternary quadric, we have the following tables: — 



Covariant and other Tables, Nos. 51 to 56 (a ternary quadric). 

No. 51. 
The quadric is represented by 

which means 

cwj» 4- 6y' + c^ + %fyz + igzx 4- 2Jucy . 

No. 52. 

The first derived functions (omitting the factor 2) are — 

(a, A, gjx, y, t), 
(A, 6, fjx, y, z\ 

No. 53. 
The operators which reduce a covariant to zero are 

( h, 6, 2/ $3^, df, dc)'-zdy, 

( a, 2A, g^Ky 9b, d/)-ydg, 

( g, 2/ c$3a, 3*, 3/)-y3„ 

( a, A, 2g'$dg, 3/, 3c) --^3,, 

(2A, b, f^da, 3a, 3^)-a3y. 

No. 54. 
The evector is 

(3a. 3ft, 3c, 3/. dgy dk\^, 17, f)'. 



144] 



A THIRD MEMOIR UPON QUANTICS. 



323 



The discriminant is 



which is equal to 



No. 55. 

a, h, g 
h, b, f 

abo - af* - b^ - ch* + 2fgh. 



No. 56. 
The reciprocant is 

f, a, A, g 
Vf K b, f 

which is equal to 

(6c-/», ca-g\ ah-h^ gh-^af, hf^bg, fg - ch\l ri, ^. 

The discriminant is, it will be noticed, the same function as the Hessian. The reci- 
procant is the evectant of the discriminant. The covariants are the quadric itself and 
the discriminant; the reciprocant is the only contravariant. 

Next, for a ternary cubic, we have the following Tables : 



Covariant and other Tables, Nos. 57 to 70 (a ternary cubic). 

No. 57. 

The cubic \a U^ 

(a, 6, c, /, g, A, i, j, k, Z$a?, y, zf, 

which means — 

aa^ + bf-¥c2^ + Sfy^z + Sgz^x + Sha^ 4- %-&• + Bjza^ + 3fcry" + 6lxyz, 



No. 58. 

The first derived functions (omitting the factor S) are 

(a, k, g, I, j, hjx, y, z)\ 

{K b, %, f, I, kjx, y, zf, 

(jy /, c , i, g, I '$x, y, z)\ 



41—2 



324 



A THIRD MEMOIR UPON QUANTICS. 



[144 



The second derived functions (omitting the factor 6) are 



(a. 


h. 


j $a;, 


y> 


^). 


(k. 


b. 


fl^. 


y> 


A 


iff. 


• 


c l^x, 


y. 


A 


(I. 


/. 


ijix, 


y> 


'). 


a 


I. 


gl'^' 


y> 


'). 


(h, 


k. 


I T§ix, 


y. 


')■ 



No. 59. 
The operators which reduce a covariant to zero are 



(j. 


3/. 


c, 


2t, 


9. 


(o. 


k. 


3«7. 


ii, 


2j. 


(3A. 


b, 


• 


/ 


2«, 


(A. 


b, 


3i-. 


2/. 


2i. 


(3j, 


/. 


c, 


• 

t, 


2*7. 


( a, 


Sk, 


S'. 


21, 


i. 


The evector is 











2Z$3a, 96, 9<, 9/, 9(, 
A$9j, 9/, 9c, 9,-, dg, 
2kJida, 9*, 9^ 



ff* 



9^ 9j, dk) 



^][3i> ^/» 3c> 3»» ^y» 90 — '2r9y, 

2Z$9a, 9*, 9^, 9f, 9,-, 9*) — a:9,, 

2A][9a, 9^, 9<, 9/, 9^ 9*)-y9«. 

No. 60. 



(9a, 9^, 9c, 9/, 9y, 9a, 9<, 9;, 9*, 9f3[f, 17, (;)*. 



The Hessian \a HU^ 

(a, A, j$ir, 
(A, A;, Z $ar, 

which is equal to 



No. 61. 

y, 2:), (A, A;, Z$ir, y, ^), (j, ^» 3^$^* 

y, ^), (Ar, 6, /$a:, y, ^), (Z, / i$a:, 

y, ^), (i, /, t$a?, y, 4 (jr, t, c$a?. 



y. 


^) 


y. 


^) 


y. 


z) 



' 


agk-l 


Mi-1 


cy -1 


6<*-l 


ae/-\ 


abg- 1 


bcj -1 


«cA — 1 


aW -1 


06c — 1 




aP +1 


6P +1 


c? +1 


65-^+2 


ot* +1 


flt/*/ +2 


bf +1 


«/ff-l 


oT + l 


q/i +1 




yA' +1 


/»A+1 


fg" +1 


6»y -1 


cAZ +2 


oii — 1 


c/l-1 


ail +2 


6yA-l 


hgj +1 




A/i -2 


/«-2 


fki -2 


c*» +1 


# -1 


bp +1 


c^/ +2 


eh" + 1 


y/ +2 


cAA + 1 




j»* +1 


♦*• +1 


iy +1 


/y +1 


/»■ +1 


/Hi -2 


yv +1 


^ +1 


^k-2 


/yA-3 


•\ 








/y*-2 


g'k +1 


J7W+I 


j;i*-2 


g;A + 1 


gJ^ +1 


fjl +2 










y%i + i 


j^At-2 


AS +1 


At» +1 


hij -2 


AK+1 


yA/+2 










fp -1 


j?P -1 


A^ -1 


iP -1 


iP -1 


AP -1 


Aa +2 

V* -3 


v. 




















P -2 



<l(«,y, 



:i=3 



:i=3 



dbS 



=lr5 



±6 



:t5 



±9 



144] 



A THIRD MEMOm UPON QUANTICS. 



325 



The quartinTariant w S= 



No. 62. 



abcl — 1 


/y +1 


abgi + 1 


fghl + 3 


acfk + 1 


/a;*-i 


qTg^l 


/Av -1 


afil + 1 


^? -2 


aPk -1 


i^ifc« +1 


hch^ + 1 


ghki — 1 


hfh -\ 


^A;? -2 


bgjl + 1 


AV +1 


hij" -1 


A*? -2 


qf'h -1 


ijkl +3 


cM/ + 1 


^* +1 


ciife« -1 





il6 



The sextiu variant is 7 = 



No. 63. 



a^¥<^ 


+ 1 


acpKL 


-24 


hcfhj^ 


-12 


cfhH 


- 12 


fgikV^ 


-12 


a^bcfi 


- 6 


acpjk 


-12 


hcghH 


-24 


cfh^J^ 


+ 12 


fh^3 


-12 


a^hi? 


+ 4 


acfg]^ 


-12 


hcghjk 


+ 18 


cfhjJd 


60 


fhijl? 


-12 


a^cP 


+ 4 


acfJiki 


+ 18 


bchHj 


-12 


cjr^ 


+ 24 


fim 


+ 36 


a^PP 


- 3 


cicfkJ^ 


+ 36 


hchjt' 


+ 36 


cgWl 


+ 12 


fU^ 


+ 24 


ab^cgj 


- 6 


acH^l 


-24 


hcPkl 


-24 


cgjl^ 


-12 


9"^ 


+ 8 


ahy 


+ 4 


^Poi 


-12 


hfg^hj 


+ 6 


chHkl 


+ 12 


g'hkH 


-12 


ahc'hk 


- 6 


afVk 


+ 24 


Pf9?l 


+ 12 


chijh^ 


+ 6 


ipJ^P 


-24 


ahcfgh 


+ 6 


apghi 


+ 6 


hfif 


-12 


chkU" 


-12 


gh^ki^ 


-12 


abcjjl 


+ 12 


cpgJ^ 


+ 12 


hg'hk 


-12 


i^iei^ 


+ 12 


gJJdP 


-12 


ahcgkl 


+ 12 


a/Hjl 


+ 12 


hg'hH 


+ 24 


fV 


+ 8 


gij^l 


+ 36 


abchil 


+ 12 


a/gkil 


-60 


hfh^ 


+ 12 


/yA« 


-27 


gkl* 


+ 24 


abcijk 


+ 6 


afhi?l 


+ 12 


hfikl 


+ 12 


/v* 


-12 


h^i? 


+ 8 


abcP 


-20 


af?ik 


+ 6 


hghijl 


-60 


Pghji 


+ 36 


hS*J^ 


-24 


ah/gij 


+ 18 


afU? 


-12 


bgipk 


+ 6 


phis' 


-12 


hit" 


+ 24 


ohffl 


-24 


agJ^il' 


+ 24 


bgjP 


-12 


yy^ 


-24 


hi^kl 


+ 36 


abg'ki 


-12 


aht^k 


-12 


bhPf 


+ 24 


ffhki 


+ 36 


tVifc» 


-27 


ahghi^ 


-12 


ai»kP 


+ 12 


bipP 


+ 12 


ffi^ 


-12 


ijkP 


-36 


abgU^ 


+ 36 


^cP 


+ 4 


(?h^l^ 


- 3 


fghHl 


+ 36 


Z» 


- 8 


ahi^jl 


-24 


^9T 


- 3 


cPh^j 


+ 24 


fghijk 


- 6 






04?}^ 


+ 4 


h(?h^ 


+ 4 


cfgh^k 


+ 6 


fgh^ 


-36 







±871 



The discovery of the invariants S and 
ressions were first obtained by Mr Salmon. 



T is due to Aronhold, the developed 



No. 64. 
There is an octicovariant for which we may take 

eu= d^u, dyHU, 

d^Hu, fa,' u, id^dyU, 

dyHU, idyd^u, J 3/ u, 

dHU, ifd.d^u, idyd,u. 



d,HU 



326 



A THIRD MEMOIR UPON QUANTICS. 



[144 



or else 

©.tr= ia, u, ^dy u, id, u 

id^U, a,» HU, d,dyHU, d^,HU j 
IdyU, dyd^HU, dy* HU, dJd^HU 

J a, £7, d,d,HU, d.dyHU, a,» hu 

or else, what I believe is more simple, a function 9„U, which is a linear function of 
the last-mentioned two functions. 

The relations between SU, S,U, S^U are 

-S,U+4&U=T. V^-2iS.U.HU, 
e„U+2SU=T.U*-108.U.H(T. 

I have not worked out the developed expressions. 



The cubicontravariant is PU= 



No. 65. 



/ 1 


bcl -1 


ad — 1 


ahl^l 


aek + 1 


o6i + 1 


ftc; +1 


oiy + 1 


bch+l 


a^ + 1 


abc-l 




6^+1 


a^ + 1 


a/k+l 


afg-2 


«/"-! 


V-1 


a/if +1 


bgl +1 


ot* -1 


<j/» + 1 




c/k + l 


chj + 1 


hhj + 1 


ail + 1 


6j^A-2 


c/A-2 


atA - 2 


6V -2 


eU. +1 


*»■ +1 




Pg-\ 


g'h^X 


/A>-1 


ch* -1 


6;7 +1 


ckl +1 


y« -1 


cA» -1 


<2;* -2 


cM + l 


• 


fil +1 


gjl +1 


cM +1 


/jf +2 


/hi + 3 


/i^i +3 


yv-i 


/V + 2 


/a> -1 


/gh+3 


I'A: -1 


ip -1 


j^ -1 


//*/ + 3 


j!;*-i 


yv -1 


ghk-l 


/?*-! 


g'k+2 


fjl -4 










ajk -1 


S-A* +2 


gki-l 


hH +2 


/A»-l 


ghi-\ 


y«-4 










hij - 1 


AH-1 


hi? +2 


hP -2 


/P -2 


gP -2 


Atf -4 










jP -2 


itP -2 


t? -2 


i« +3 


Ud +3 


ijl +3 


V* +3 


V. 




















P +4 



±3 



±3 



:l=8 



±7 



±7 



±7 



±7 



:fel8 



<[^, 17, 0'. 



144] 



A THIRD MEMOIR UPON QUANTIC8. 



327 



No, 66. 



The quintic contraTariant is QfT^ 



+ 1 a«6<r» + 1 


a«6V + 1 


a'^ftci - 3 


a6 V - 3 


o^c»A:- 3 


a^bcf- 3 


o^'c^- 3 


aftc'A- 3 


cibcfj + 6 


- 6 


a^cfi- 3 


a»6/» - 3 


aV + 6 


aby + 6 


ahc/g+ 3 


a'^ftt' + 6 


abcfl+ 6 


rt^c^^ + 6 


oAc^A + 6 


+ 4 


aH** + 2 


ay + 2 


ayi«- 3 


abc/h+ 3 


a6cfi + 6 


a^fH- 3 


o^ctA; + 3 


abcij + 3 


ahchi + 6 


+ 4 


ohcg^ — 6 


ai^si- 3 


a6(^A+ 3 


a6cA;^ + 6 


abgi^ — 6 


a6cA7 + 6 


ahfgi + 9 


abfi- 6 


ah<^ -30 


- 3 


ab^ + 4 


a6cM— 6 


a6c;7 + 6 


ab/ij+ 9 


ac/«Z - 12 


ahcjk + 3 


aAt«/ -12 


ac«Ar» + 6 


oW - 12 


- 3 


at?hk- 3 


a6/<7^+ 3 


ahgii + 9 


ab/gl-24: 


acfki+ 9 


afc/k; + 9 


acpk- 6 


acpj- 6 


ahgU + 36 


+ 2 


acfgh+ 3 


ab/jl+ 6 


ab^l -12 


abgki— 12 


a/V+ 3 


aft^A; - 6 


a/V - 6 


ac/^A;- 12 


o^i^J - 12 


- 3 


acfjl + 6 


abgkl+ 6 


ac/A^-24 


o^/a^- 6 


a/iH + 6 


ahqhi— 12 


a/*t/ + 6 


oc/'Ai + 9 


acfVi-\2 


+ 3 


acgkl + 6 


a6Ai^+ 6 


ac^A;-12 


a6iZ« +18 


ai'A; - 6 


06^/^ + 18 


a/*i^A; + 3 


ac/P + 18 


acfkl + 36 


+ 6 


achU + 6 


o^vA; + 3 


ctcgl^— 6 


ac/A:»- 6 


bc'h^ + 6 


aMil - 24 


6 V + 6 


ociAr^ - 24 


acih? - 12 


+ 6 


acijk + 3 


ahP -10 


acAA:t+ 9 


a/^j - 6 


bcff^ - 6 


acfhk+ 9 


bYJ - 3 


a/V + 12 


a/V/ + 12 


+ 6 


acJ^ -10 


acil» + 4 


oc*;* + 18 


a/VA;+ 24 


ici^A^ - 24 


acA:V - 12 


be/ hj- 12 


afgil - 30 


aAy+ 6 


+ 3 


«/&V + 9 


a/«A/-12 


af^gj-l^ 


a/^hi+ 3 


bcgjk + 9 


a/VA+ 3 


bcghk+ 9 


a/i«J + 3 


afgki - 30 


-10 


affl - 12 


a/y*- 6 


afg'k + 24 


o/«P + 6 


6c/«>*-12 


«/!;7+ 6 


&cA«t - 6 


agki^ + 24 


a/At«+ 6 


+ 9 


a^ki — 6 


afyk^- 6 


afghi + 6 


qfkil - 30 


bcjP +18 


o/^A;; - 30 


6cAP + 18 


aAt' - 6 


afiP - 18 


-12 


(ighi^ — 6 


a/hki+ 9 


o/i^P + 12 


oAr^i^ + 12 


*/i^!; + 3 


a/hil+ 12 


bcjkl - 24 


a?P + 6 


ai'A/ + 12 


- 6 


agi^ +18 


o^^P + 18 


afijl + 12 


l^gP - 3 


bg'k - 6 


fl/t; A; + 6 


ft/^A+ 3 


bcghj + 9 


bcgh^ - 12 


- 6 


ai?jl -12 


o*A:»; - 12 


flw/^iY - 30 


bchH - 12 


6/Ai + 24 


a/P - 6 


Vk;'^ + 12 


6c// -12 


bchjl + 36 


+ 18 


6cf + 4 


^/ + 2 


ahn + 6 


fccA/A: + 9 


V^ + 6 


a^A:^ +24 


b/if - 18 


bg% - 6 


6c/A - 12 


12 


V/ - 3 


6cA» + 4 


ai^'A; + 3 


b/g/ij+ 6 


6^^; - 30 


aAi2Ar-18 


^^iT'A:/ + 6 


Vi^ + 6 


bfgp + 6 


+ 2 


c»A» + 2 


b/hj'- 6 


aiP - 6 


ftj5^^ + 6 


biY +12 


aikP + 12 


bghil - 30 


bgij' + 3 


6i^A/ + 12 


-12 


cfhp- 6 


6^A»^-12 


bchj' - 6 


bg^hk- IS 


c«AAr» - 3 


6cAV - 6 


6^/;A: + 6 


(T^A^A - 3 


bg^jk + 6 


- 6 


c^A«/ - 12 


bghjk + 9 


6^Ai+ 3 


bghH + 24 


cj^hj + 24 


b/f - 6 


bgP - 6 


c/gh^+ 3 


bghij - 30 


- 6 


cghjk-k- 9 


&A«t; - 6 


b^l + 6 


bghP + 12 


c/ghk+ 6 


6(/W + 12 


bhi^j +24 


r/^i^ - 30 


6^'/« -18 


+ 9 


c/Ay — 6 


bhjP + 18 


6v» - 6 


6fiyAr; + 12 


c/A^ - 18 


6^A;7 - 30 


6i;P +12 


c/j'k + 24 


bifl +12 


+ 18 


chjl^ +18 


6/)W - 12 


cm + 24 


bhijl - 30 


c/hP + 12 


bgfk + 3 


c/Vi'^ + 12 


c^AA;+ 12 


c/A«/ + 12 


-12 


c/A:Z -12 


c/i^k^ - 3 


c^A«A;+ 3 


6i/«>fc + 3 


ciO'A;^ - 30 


6Aty« + 24 


c/hkl - 30 


cgjk^ - 18 


c/A;A - 30 


- 6 


//Ai+ 3 


/»/i!/ + 12 


chH - 6 


bjP - 6 


cg^l + 6 


b/P + 6 


c/j/(^ + 24 


chHl + 6 


cgrAA* + 6 


+ 12 


A;"'^ + 6 


/^/*»A:+ 3 


cA«Z» + 6 


c/h^k+ 3 


cAiAr^ + 12 


r/A» - 6 


c^A:* - 6 


cAyA + 6 


chHk + 6 


+ 3 


fif - 6 


/hH - 6 


chjkl - 30 


cAA:*/ + 6 


cij^ + 3 


cA«A:; + 6 


cAiA:« + 3 


cAZ» - 6 


chkP - 18 


+ 6 


^iW: - 6 


/A«^ + 6 


cf^ +12 


cjk" - 6 


ckP - 6 


chjk" + 3 


ck'P + 6 


9 A/2 +12 


cjm +12 


+ 6 


g'hH +12 


fhjkl-ZO 


/y +12 


/V*' - 27 


/yA - 27 


/V- 6 


/T +12 


/ V - 6 


/V^i + 18 


-30 


i^AZ« + 6 


ffl^ +12 


//A« - 27 


/!;"'^ - 6 


Pgil + 18 


/^A«/ + 18 


PiUk -12 


/^A/ + 18 


Pjn -24 


+ 6 


g'jkl + 6 


i^AJfe«Z+ 6 


/«;•«* -12 


/%7 + 18 


Z*^/* - 6 


/^A;*- 3 


Pghl + 18 


fg'jk -12 


/</»M + 18 


+ 3 


^Av7 - 30 


9i1^ - 6 


/^/i;7 + 36 


/ghkl + 36 


/v'*^ + 18 


/hHj -12 


/Va>- - 12 


/^Ai;- 3 


fghH + 18 


- 6 


gipk + 3 


A«tA:Z + 6 


/hip - 12 


/^•^-12 


fqhil + 36 


/Ai^ - 6 


PP -24 


/^•^ - 6 


fghP - 54 


+ 12 


<Z;7» - 6 


Ai/jfe« + 3 


;!rP -24 


/A«i/ + 18 


/W^- 3 


ffkl + 18 


//A? - 6 


/(;•'/ + 18 


/7iA/ - 12 


- 6 


h%y +12 


MP - 6 


^hkl + 18 


/Ai/A- 3 


fgl? -18 


i^AA:> - 6 


fghik-Z 


g'k' +12 


/Ai;7 -12 


+ 6 


i7/» + 6 


iA«P + 6 


!/!;^ - 6 


/A/» -18 


yAt^i -12 


gh^ki- 12 


fgkP - 6 


g%ki-\2 


/i/*A+18 








ghHl + 18 


j5^p - 6 


/i/P - 6 


ghkP - 6 


fhS" - 6 


fkP -24 


/;/» +48 








ghijk - 3 


^F +12 


fiei - 6 


gjm + 18 


/AiP - 6 


^A«i» - 6 


fkH -24 








^/i^» -18 


(^/lA^^i - 12 


ghk^ - 12 


A\^ + 12 


fijkl + 36 


ghiP - 6 


ghkil- 12 








i^'/t/^ - 6 


gk^P -24 


i^>ti7» - 6 


hHP -24 


/^ + 12 


^(/A/ + 36 


gij^ + 18 








AVy - 6 


/i«A:i« - 6 


AV + 12 


A/^ + 12 


giMl + 18 


gP + 12 


^A/* +48 








hip - 6 


fikiP - 6 


Ai«P -24 


hijkl + 36 


Ai^A;; + 18 


At^/ +18 


AW -24 








V*A;Z +18 


t/A:*/ +18 


i^jkl +18 


ifJ^ -27 


t^A* -27 


i^j'k -27 


hiP +48 








p + 12 


kl* + 12 


t7* + 12 


JAjP -18 


ikP -18 


iA/» - 18 


Ai!/A + 18 
ijkP -54 
Z» -24 



'\ 



^{t V (r 



dbl45 



14o 



i:282 



=b282 



:i^282 



:1=282 



:t282 



282 



db486 



328 



A THIRD MEMOIR UPON QUANTICS, 



[144 



No. 67. 
The reciprocant is FU={*^^, 17, f)' = 



^ 



W +1 
bc/i - 6 
W» +4 

cP +4 



aV +1 
acgj— 6 
o^ +4 
cf +4 

pV-3 



f^. 



a»6» +1 
abhk — 6 
oifc* +4 
6A» +4 
h^Jc" -3 



^r. 



r»^ 



9 * 



- 6 
aegh + 6 



ag'l 
chf 

if 



+ 12 
+ 18 
-24 
-12 
+ 6 
+ 12 
-12 



abfh 

ahkl 

afie 

hhH 

bhjk 

fh^k 

hiei 



- 6 
+ 6 

+ 12 
-12 
-24 
+ 18 
+ 6 
+ 12 
-12 



iS- 



hi?k 
hcfg 
bcil 

qfH 



- 6 
+ 6 

+ 12 
-12 
-24 



c/H + 18 

ft'l +12 
« - 12 



iji*. 



a V - 6 
abhl + 12 
abjk + 6 
a/hk + 18 



ak'l 

bhy 



-24 

-12 
-12 
+ 12 

+ 6 



if. 



l^cg 
btifl 
bcik 

hfgi 

bH'l 
cPk 

P9 
A^k 



- 6 
+ 12 
+ 6 
+ 18 
-24 
-12 
-12 
+ 12 
+ 6 



fif". 



ac»A - 6 
ciegl + 12 
acij + 6 
agH -12 
cghj + 18 
cpi -24 
^A -12 
9^1 +12 

^i" + 6 



±9 



9 



9 



54 



54 



54 



54 



db54 



54 



n*i*. 


W 


W. 


i?»f*. 


i*e*. 


ev^> 


T'f. 


f*?. 


?»i». 


a V + 6 


o^V + 6 


6c«A + 6 


a»6» + 6 


6»c/ + 6 


ac'k 


+ 6 


a'be - 2 


a6«c - 2 


oW - 2 


aV + 9 


ab/l - 12 


6c^/ -12 


ay + 9 


by + 9 


acfg 


- 6 


aVi - 18 


a6/t + 6 


ac/» + 6 


achl - 12 


abik — 6 


bcij - 6 


a5^/* — 6 


fcc/A - 6 


acil 


-12 


abgj + 6 


a/» - 4 


at' - 4 


oc; A; - 6 


a/»A; + 12 


6^ +12 


abjl -12 


6dW -12 


agp 


+ 12 


achk + 6 


6«a; -18 


ftca; + 6 


a/gj -18 


^i" + 9 


c»A:» + 9 


a/A/ - 36 


b/gl -36 


c»A« 


+ 9 


afgh + 18 


6cA)k+ 6 


V - 4 


o^A; +12 


6/A; - 18 


c/y +12 


a/jk - 18 


b/ij -18 


cJP 


+ 12 


afjl +36 


bfgh + 18 


c'Ait -18 


aghi - 18 


bghk - 18 


(2/5^A; - 18 


agk" +12 


6^ -18 


cghl 


-36 


agkl - 48 


^^gf/ +36 


c/flrA + 18 


a^^ +48 


bhH +12 


c/hi - 18 


atAifc - 18 


6Ai» +12 


cgjk 


-18 


ahU +36 


bgld + 36 


cfjl -48 


aijl -36 


6AZ« +48 


c//» +48 


oA^ +48 


biP +iS 


chij 


-18 


aijk +18 


6AiZ - 48 


<^A^ +36 


cAV +12 


6;A^ -36 


o^Z -36 


bkp +12 


c/*» +12 


cjP 


+ 48 


a/* -32 


bijk +18 


cijk +18 


or +12 


/W - 3 


/y - 3 


^A'A: - 6 


/!/ +12 


m 


- 6 


6;' - 4 


6/" -32 


cAt/ +36 


i^A» - 3 


/AA;Z - 24 


/gil -24 


A'j +36 


/^gk + 36 


fk 


+ 12 


cA» - 4 


cP - 4 


c/» -32 


5rA;7 -24 


^k" +36 


/»!; - 6 


A«t +12 


/•Af - 6 


fhi 


+ 36 


/y -36 


/»AZ + 12 


/^Z +12 


Sfk - 6 


^/fc* +12 


gi'k +36 


h^P -12 


PP -12 


g'P 


-12 


ghH +12 


Pjk - 36 


^V +12 


Atf +36 


hik" - 6 


Ai» + 12 


A;;M -24 


/a/ - 24 


gijl 


-24 


(^Ajife + 12 


/i7ife» -36 


g'ki -36 


fP -12 


*»Z» -12 


*•/« -12 


/*» - 3 


Pk" - 3 


»y 


- 3 


AV -36 

hjP +24 
y»« +12 


fhki +12 
y^ +24 
tib"Z +12 


^At« -36 
^ +24 
Pjl +12 



135 



135 



:i^l35 



135 



dbl85 



185 



180 



11x180 



rfelSO 



U] 



A THIRD MEMOIR UPON QUANTIC8. 



329 



12 
30 
12 
24 
18 
12 
24 
24 
12 
18 
60 
12 
66 
12 
48 
60 



V*^' 



ac/j 
acgk 
achi 

agtl 
ai?j 

egh^ 
chjl 

cpk 

fhl 

fjk 
ghij 

jHl 



-12 
-12 
+ 30 
-24 
+ 24 
+ 12 
-18 
-18 
+ 12 
+ 24 
-12 
+ 60 
-12 
-66 
-48 
+ 60 



i^fi?. 



abgk 

abhi 

abij 

abP 

aph 

afkl 

aiJ^ 

hgh^ 

hhjl 

bj^k 

fhH 

fhjk 

gW 
h^ki 
hkP 



-12 
-12 
+ 30 
-24 
-18 
+ 12 
+ 24 
+ 24 
+ 12 
-18 
+ 60 
-66 
-12 
-12 
-48 
+ 60 



wr. 



cibci 

aep 

a/t^ 

hcgh 

hcjl 

hfl 

igij 
cfhl 

cfjk 

cg^ 
chik 
ckP 

fgh% 

gikl 

hn 

i^jk 



+ 6 
-12 
+ 6 
-24 
+ 36 
+ 12 
-30 
-60 
+ 12 
-36 
+ 54 
+ 24 
-24 
+ 60 
+ 78 
-96 
+ 72 
-12 
-48 
-66 
+ 48 



7»r»^ 



abcQ 

abg^ 

acfh 

ackl 

afgl 

a/ij 

ahi^ 

agik 

ail^ 

chH 
chjk 
fghj 

m 

g'hk 

ghH 

ghJI? 

gjkl 

hijl 

ij^k 



+ 6 
-12 
-24 
+ 36 
-60 
+ 54 
-36 
+ 12 
+ 24 
+ 6 
+ 12 
-30 
+ 78 
-48 
-24 
+ 60 
-96 
+ 72 
-12 
-66 
+ 48 



f*ft7». 



abck 

ah/g 

Ml 

api 

a/ik 

hch^ 

hghl 
bgjk 
bhij 
bp 

cm 

fghJc 

fhH 

fhJ^ 

fi^i 
gm 

hikl 



+ 6 
-24 
+ 36 
+ 12 
-30 
-12 
-36 
-60 
+ 54 
+ 12 
+ 24 
+ 6 
+ 60 
+ 78 
-24 
-96 
-12 
-48 
+ 72 
-66 
+ 48 



^17^'. 



abcf 

abi^ 

afH 

bchl 

bcjk 

bfk 
bghi 

bffP 
bijl 
e/hk 

cm 
Ml 

fhil 

fijk 

fP 
giJ^ 

hi^k 

ikP 



+ 6 
-12 
+ 6 
+ 36 
-24 
+ 54 
-36 
+ 12 
+ 24 
-60 
-30 
+ 12 
-66 
-48 
-12 
+ 72 
+ 78 
+ 48 
+ 60 
-24 
-96 



iy»rt«. 



ahcg 
acfl 
acik 

ofgi 
aiH 

bcp 

Vi 

cghJc 
chH 
chP 
cjkl 

fail 
g'kl 

ghil 
gijk 

hiy 



+ 6 
+ 36 
-24 
-30 
+ 12 
-12 
+ 6 
+ 12 
+ 54 
-36 
+ 24 
-60 
-66 
+ 72 
-24 
-48 
-12 
+ 78 
+ 48 
+ 60 
-96 



f'^^. 



ahch 
abgl 
abij 

a/gk 
afhi 
afP 
aikl 

hghj 

bfl 
ch^k 

fhjl 

m 

ghkl 

h^il 
hijk 
hP 
jkP 



+ 6 
+ 36 
-24 
-12 
-36 
+ 12 
+ 54 
+ 24 
-60 
-30 
+ 12 
+ 6 
-66 
-12 
+ 60 
+ 72 
-24 
-48 
+ 78 
+ 48 
-96 



wr». 



abcl 
aJbgi 
acfk 

oPg 

afxl 
w?k 
bchj 
bgVi 
bgjl 
bip 

cfli^ 
chkl 

\fgh.l 

/jp 

g'i^ 

ghkl 

gkP 

h^? 

hiP 

ijkl 

I*. 



}22 



db222 



=1=222 



:i=408 



:Jb408 



±498 



±408 



408 



408 



24 
6 

6 
30 
48 
30 
6 
30 
48 
30 
30 
48 
30 
24 
+ 108 
-114 

- 114 
+ 24 
+ 24 
-114 
+ 24 
+ 24 
+ 24 
+ 108 

- 48 

±558 



+ 
+ 

+ 

+ 
+ 

+ 

+ 

+ 

+ 
+ 



The preceding Tables contain the complete system [not so] of the covariants and 
ntmvariants of the ternary cubic, i.e. the covariants are the cubic itself U, the 
lartinvariant 8, the sextinvariant T, the Hessian HU, and an octicovariant, say SU; 
e contravariants are the cubicontravariant PU, the quinticontravariant QUy and 
e reciprocant FU. 

The contravariants are all of them evectants, viz. PU is the evectant o{ 8, QU 
the evectant of T, and the reciprocant FU is the evectant of QU, or what is the 
me thing, the second evectant of T. 

The discriminant is a rational and integral function of the two invariants; repre- 
Qting it by E, we have i2 = 64 8^-1^. 

If we combine U and HU by arbitrary multipliers, say a and 6)8, so as to form 
e sum aU +6/3HU, this is a cubic, and the question arises, to find the covariants 
d contravariants of this cubic : the results are given in the following Table : 



7-^6^HU 
{aU-^6/3HU) 



aU-^e^HU. 

(0, 2S, T, 
+ (1, 0,-125, 



No. 68. 

8S«3[a, /3yu 



C. II. 



42 



aso 



A THIRD MEMOIR UPON QUANTIOS. 



144 



P(aU+6^HU)= (1. 0, 125, iT^a, ^yPU 

4(0,1. 0, - 'iSlia. ^y QU. 
(0, 60.Sf, SOT, 0,-1 20T8. - 242* + 5765* \a, fiy P U 

+ (1, 0, 0, lor, 2405*, 2^T8\a, ^yqu. 

(5, T, 245', 4^5, 2*-485»3^a, ^y. 
{T, 965", 6075, 202*. 2402*5*. - 482*5 + 46085*. - 82* + 5762'S»'J[a, jS/. 
[(1,0, -245,-82', -4&S'\t, ^y\*R. 
(1,0. -245, -87, - 4»S''$_a, 0y FU 

+ (0. 24, 0, 0, -isT^cL, fiy(PUy 

+ (0. 0. 24, 0, 9QS\a, fiyPU.QU 

+(0, 0, 0. 8. 03[a, fiy.{QUy. 

We have, in like manner, for the covariants and contravariants of the cubic 
GttPU+l3QU, the following Table: 



5 (aU + 6fiHU) = 
T(aU+6fiHU) = 
R(aU+Q^HU) = 
F(aU+6fiHU) = 



No. 69. 



6(xPU+fiQU =fi<iPU + PQU. 

H(6aPU + fiQU) = (-2T. 485*, 182'5, 2* + 165»'5a, ^yPU 

+ (85, T, - 85«, - T8 \<i, ^y Q U. 
P (6aPU+ ffQU) = (325*, 122'5, 2* + 325', 42'5'3^a, /9)» U 

+ (47, 965'. 122'5, 2* - 325* 3[«, ^S)* HU. 

Q(QaPU + 0QU)= r + 3842'5», 

+ 1202*5 + 7680 5*, 
+ 102* +320075*, 
+ 4802*5», 
+ 302*5, 
1^+ 12^ - 242*5* + 6125" 

+ (- 242* +4608 5>, "^ 
+ 19202* 5*. 
+ 4802*5, 

+ 302'* + 19202'5», 
+ 1202"5» + 7680 5*, 
62"5 + 76875* 



6 a, ^yU 



) 



a, fiy HU. 



44] 



A THIKD MEMOIR UPON QUANTIC8. 



331 



8(6aPU + ^QU) = 



T(6aPU+l3QU) = 



(+ ir» +192 S", ' 

+ 128TS\ 

+ 18T*S + 384 S\ k a, fi)\ 

+ IT* + 64^/8", 
t+ 5r»S«- 64 -Sf" J 

'- 8r» + 4608rS», 
+ 1920r»iS» + 73728 /S*. 



"i 



+ 360r»<Sf + 384002'/S', 

+ 2or* + sgeor'/S", 
+ 84or'/S»+ lesoTS', 

+ S6T*S + SSiPS* + 24576 S', 
{+ 12" - 40r'/S»+ 2560rS* ; 

+ /3QL0 = [(48&', Sr, -965*, - 247/8, - 2" - 16/Sf "go, /3)']»i?. 

f/8Qfr) = ( 192S, 322* ,-384 S' ,- 96T8 

+ ( , 512 iS», 1922* /Sf , 24r'/Sr 

+ (1344S', 3522'«S, 24r'-1152S», -2882'S» 
+ (48 2'. , 2882' -S , 247' + 1536/8*, 



6 a, /3)'. 



47' -64fi^3[a, /3)*eir 
2'' 5a, /9y. U* 

20T'8 + 64/S*3ia, /9)« U. HU 



The tables for the ternary cubic become much more simple if we suppose that 
16 cubic is expressed in Hesse's canonical form; we have then the following 
ible: 



U 

S 

T 

R 

HU 

SU 



e.u = 



No. 70. 

^ + y* + •** + Qlxyt. 

-l+l*. 

1 _ 20? - 81*. 

- (1 + 8P)». 

I* (x* + y* + z*) - (1 + 21') xyz. 

a + 8J*y {fi> + z'a^+ ai'y*) 

+ (-9l*)(x' + y* + ^y 

+ {-2l- ol* - 20r) (a;» + y» + ^») xyz 

+ (- 151* - 781' + 121*) ai'y'z*. 

4 (1 + 8l*y (y*!* + 1^*0^ + a?y*) 

+ (-l-W-4>l*)(ai> + y* + z»y 

+ (4/ + lOOi* + 112P) (a!» + y» + «») xyz 

+ (48f + 552P + 48f) !d*y*z*. 



42—2 



332 



A THIRD MEMOm UPON QUANTICS. 



[144 



^,,U= - 2 (1 + Sl'fifz' + 2^0^-^- a^f) 

+ (i-ioz»)(^ + y' + -8*)» 

+ (6i - 180Z* + 96^) (a^ + y» + ^) ayyz 
+ (6f» - 624P - 192i«) a^fz\ 

PU =-Z(p + i?* + (r) + (-l + 4?)^7C. 

QU =(l-10P)(f* + if + (r)-6?(5 + 4i»)fi;?. 

FU = - 4 (1 + 8P)(i;»?» + ?»p + fi7») 

- 24Z (1 -f 2f») fi7»^, 
to which it is proper to join the following transformed expressions for ©CT, ©,i/, B^^IT, 
viz. 8J7 = (1 + 8Z»y (y»^ + ^a^ + a;»y») 

+ (2Z - 5Z*) 17 . iTIT 

+ (-3i» )(irco. 

e,J7 = 4 (1 + 8Z»)» (y»^ + -?»a^ + «»y») 

+ (-16i + 4/* )U.HU 
+ (-12i« )(^^'. 

e,,ir= -2(1 + 8P)«(y»^ + -?»a:» + a^y») 

+ (6Z )i;'.irfr 

+ (6Z» )(HU)\ 

The last preceding table affords a complete solution of the problem, to reduce a 
ternary cubic to its canonical form. 



[I add to the present Memoir, in the notation hereof (a, 6, c, /, g^ A, i, j, it, V^x, y, zf 
for the ternary cubic, some formulae originally contained in the paper "On Homo- 
geneous Functions of the third order with three variables," (1846), but which on account 
of the difference of notation were omitted from the reprint, 35, of that paper. 

Representing the determinant 



cuc + hy + jz, 
hx + ki/ + bz, 
jx + iy + gz, 



hx + ky + bz, jx +li/ + gz, f 
kxfby+fz, Ix +Jy + tz, tf 
Ix -^rfy + iz, gx + %y + cz, f 



A THIRD MEMOIR UPON QUANTICS. 



333 



(A, B, C, F, 0, E^x, y, zf 
alues of -4, B, (7, F, 0, H (equations (10) of 35) are 



jover writing 



lat 



B 



G 



H 



^ 



Cf 



^ 



- p 


• bi 

- r 


- i* 


be 
-fi 


fg 

+ ek 
-2U 


ki 
+ bg 
-2fl 


ag 
- J' 


hi 

- p 


- f 


• ■ 

V 
+ eh 

-2gl 


ea 

- SO 


gh 

+ M 

-2jl 


ok 

- A» 


bh 


- P 


¥ 

+ bj 

-2kl 


+ a/ 
-2hl 


ah 

- hk 


2M 
-2al 


2kl 
-2V" 


2gl 
-2t>- 


P 

+ gk 
- hi 


gh 

— ai 


- «/ 


2hl 
-2jk 


2/k 
-2U 


2*; 
-2/& 


ki 

- hg 


P 
+ hi 

-Jj 

- gi^ 


¥ 

- y 


2jl 
-2gh 


2/7 
-2ki 


2^ 
-2el 


f9 

- ek 


• • 

- cA 


p 

- gk 

- hi 



FU^' 



a, k, g, I, j, h, 

h, h, i, f, I, k, 

j, f, e, i, g, I, 

2f . . . K V 

A, B, C, F, G, H 






FU = A& + Eo + Cc + 2Fi+ 2(?g + afHi, 



334 



A THIRD MEMOIR UPON QUAKTICS. 



[14 



then the values of a, b, c, f, g, h (equations (13) of 35) are 










(?e 



(W 



eni 



w 



^c 






2c/ 
-2t« 


2bi 
-2/» 


- 6c 








2q; 
-2g> 




2a5r 
-2/ 





si 

— ca 





2hh 
-2P 


2aA; 
-2A» 










Aii; 
- ab 


-6y 
-2cA 





4A; 
-4a/ 


2/ 
-2a^ 


3at 
^2jl 


ca 

- 93 


Afk 

-46/ 


Shi 
-6; A; 
-2a/ 





a6 


2A:« 
-26A 


-2kl 

- ¥ 





4^i 
-4c/ 


8// 
-26^ 


3cA; 
-2i/ 


6c 


2»' 
-2c/ 


Ski 
-26; 


Ahj 
-4a/ 





2A» 
-2a^ 


ab 

- hk 


3a/ 
-2« 





8i/ 
-2cA; 


4/A; 
-46/ 


36<7 
-2/1 
- ki 

ca 


2/« 
-26* 


6e 
- f* 


Agi 
-4c/ 





Sjl 

-Qgh 

-2ai 


3cA 
-2gl 

• m 

- « 


2p» 
-2c; 


6Ai 
+ 6/i 
-4(7A; 
-8/» 


2asr 
-2/ 


2aA; 
-2A* 


Aal 
-Ahj 


+ 2AZ 
-3a/ 


+ 2j/ 
-3a» 


26t 
-2/» 


6^' 
-4A» 


26A 


¥ 
+ 2kl 

-3bj 


46/ 
-4/* 


Id 

+ 2/1 
-Sbg 


2c/ 
-2i^ 


2c;- 
-2<^ 


6gk 
-SP 


m m 

V 

+ 2gl 

-3ch 


+ 2t/ 
-3ck 


id 
-igi 


2/i 
-26c 


4cA 
+ 4(7/ 
-8t; 


46; 
+ 4A:/ 
-8/t/- 


AP 
+ 2hi 

-Sgk 


-6/1 
- hg 


ifg 

- dc 


4db 

4-4t/ 


-2ca 


4a/ 
+ 4A/ 
-8iA; 


7gh 

- 6jl 

— ai 


4P 

+ 2y* 
-8A» 


7v 
-6gl 
— ch 

• 


46(7 
+ 4// 
-8A:t 


Aai 
+ Ajl 

+ Sgh 


2M 
-2a6 


7jk 

-ehi 
- ¥ 


7A/ 


4P 
+ 2y* 
+ 2Ai 
-8;5f 



144] 



A THIRD MEMOIR UPON QUANTIC8. 



335 



Also if the discriminsat be written 



K(U) = 



a 


k 


9 


I 


• 

3 


h 


h 


b 


■ 

I 


f 


I 


k 


• 


• 


c 


• 
X 


9 


I 


^ 


m 


05 


% 


% 


n 


n 


iS 


I 


s 


% 


m 


aj 


I 


OD 


{ 


as 


% 



then the values of a, 33, ®, Jp, ffi, |^, I, 2I» ^> 'I (equations (20) of 35) are 

a = a% + 2hjl " aP - gh^ - j^k, 

33 = 6iA + 2/ikf - 6P - A/« - Jfc^i, 

(!C=(2/' +25n7 - cfi - /^r^ - i^;, 

3JF = 6cA + K; - cA:» +2gfk-2bgl+fP ^pj - /lA, 

3ffi = ca/ + cjk - ai« + 2A5ri - 2cAi + 5rP -^r'A? - 53/; 

3|^ = a65r+ aJfci - 6f +2fhj ''2afl + hl^ -hH - AJfc^r, 

3 I = 6c; + c/t - 65r» + 2% - 2ckl -hjl^ - i»A - fij, 

33J = CO* + oflr/- cA> + 2ijh - 2aa + kl^ -ff - a/Jfc, 

3!a = aW + bhg" ap + 2j*/ - 2hjl^il^ -k^g - hki, 

61 = ofcc + 3^A + Sijk +2/* - a/i- bgj-chk - 2Z5fA; - 2iAi - 2ljj, 



The equation JST ( IT') = i2 = 64iS* — T* would however afford a perhaps easier formula for 
the calculation of the discriminant.] 



336 



[145 



145. 



A MEMOIR UPON CAUSTICS. 



[From the Philosophical Transactions of the Royal Society of London, vol. CXLVII. for the 

year 1857, pp. 273—312. Received May 1,— Read May 8, 1856.] 



The following memoir contains little or nothing that can be considered new in 
principle; the object of it is to collect together the principal results relating to caustics 
in piano, the reflecting or refi::acting curve being a right line or a circle, and to 
discuss, with more care than appears to have been hitherto bestowed upon the subject, 
some of the more remarkable cases. The memoir contains in particular researches 
relating to the caustic by refraction of a circle for parallel rays, the caustic by 
reflexion of a circle for rays proceeding from a point, and the caustic by refiuction 
of a circle for rays proceeding from a point; the result in the last case is not 
worked out, but it is shown how the equation in rectangular coordinates is to be 
obtained by equating to zero the discriminant of a rational and integral function of 
the sixth degree. The memoir treats also of the secondary caustic, or orthogonal 
trajectory of the reflected or refracted rays, in the general case of a reflecting or 
refracting circle and rays proceeding from a point; the curve in question, or rather 
a secondary caustic, is, as is well known, the Oval of Descartes or 'Cartesian': the 
equation is discussed by a method which gives rise to some forms of the curve which 
appear to have escaped the notice of geometers. By considering the caustic as the 
evolute of the secondary caustic, it is shown that the caustic, in the general case of 
a reflecting or refracting circle and rays proceeding from a point, is a curve of the 
sixth class only. The concluding part of the memoir treats of the curve which, when 
the incident rays are parallel, must be taken for the secondary caustic in the place 
of the Cartesian, which, for the particular case in question, passes off to infinity. In 
the course of the memoir, I reproduce a theorem first given, I believe, by me in the 
Philosophical Magazine, viz. that there are six different systems of a radiant point 



145] A MEMOIR UPON CAUSTICS. 337 

and refracting circle which give rise to identically the same caustic, [see post, xxviii]. 
The memoir is divided into sections, each of which is to a considerable extent in- 
telligible by itself, and the subject of each section is for the most part explained 
by the introductory paragraph or paragraphs. 

I. 

Consider a ray of light reflected or refracted at a curve, and suppose that ^, 17 
are the coordinates of a point Q on the incident ray, a, fi the coordinates of the 
point Q of incidence upon the reflecting or refracting curve, a, b the coordinates of 
a point N upon the normal at the point of incidence, x, y the coordinates of a 
point q on the reflected or refracted ray. 

Write for shortness, 

(6-y3)(f-a)-(a-a)(,,-^)= VQQN, 
(a - a) (f - a) + (6 - /S) (i, - /S) = U^QN, 

then ^QGN is equal to twice the area of the triangle QON, and if f, 17 instead of 
being the coordinates of a point Q on the incident ray were current coordinates, the 
equation VQGN=0 would be the equation of the line through the points G and N, 
Le. of the normal at the point of incidence ;* and in like manner the equation 
□QG^i\r = would be the equation of the line through G perpendicular to the line 
through the points G and -AT, ie. of the tangent at the point of incidence. 

We have 



and therefore identically. 



NG =(a-a)» + (6-)8)«, 

W 'QG'='^QG^+ dqgn\ 



Suppose for a moment that <f> is the angle of incidence and <f>' the angle of reflexion 
or refraction; and let fi be the index of refraction (in the case of reflexion /Lt = — 1)^ 
then writing 

(6-y8)(a;-a)-(a-a)(y-;8)=V5rGi\r, 



and 



we have 



56« = (a:-a)« + (y-i8)^ 



. ^ VQGN . ^, VqGN 
''''''' = NGTGQ' ^^^*=^GTG^' 

and substituting these values in the equation 

sin* (fy-fi^ sin* 0' = 0, 
C. II. 



43 



338 A MEMOIR UPON CAUSTICS. [l45 

we obtain 

^* V^GN' - fi' Q^VqGN^ = 0, 

an equation which is rational of the second order in x, y, the coordinates of a point 
q on the refracted ray; this equation must therefore contain, as a factor, the equation 
of the refracted ray; the other factor gives the equation of a line equally inclined 
to, but on the opposite side of the normal; this line (which of course has no physical 
existence) may be termed the false refracted ray. The caustic is geometricallt/ the 
envelope of the pair of rays, and for finding the equation of the caustic it is 
obviously convenient to take the equation of the two rays conjointly in the form 
under which such equation has just been found, without attempting to break the 
equation up into its linear factors. 

It is however interesting to see how the resolution of the equation may be 

effected; for this purpose multiply the equation by Nffi, then reducing by means of 
a previous formula, the equation becomes 

C^^GN' + a^Qlt)VQGN^ - A'^W^^ + aQGN')'7^N' = 0, 
which is equivalent to 



and the factors are 

Vj(?iV\//A«DQGy + (M— l)VQGi7" TDqGN.VQGN^O; 

it is in fact easy to see that these equations represent lines passing through the 
point G and inclined to GN at angles ± <t>\ where 0' is given by the equations 

sm<f> = fi sin <f>\ 

and there is no difficulty in distinguishing in any particular case between the refracted 
ray and the false refracted ray. 

In the case of reflexion /4* = — I, and the equations become 

^qGN. DQGN+ UqGN . VQGN^O; 
the equation 

VqGN, DQGN-OqGN. VQGN^O 

is obviously that of the incident ray, which is what the false refi*acted ray becomes 
in the case of reflexion ; and the equation 

VqGN . DQGN-hDqGN. S/QGN = 
is that of the reflected ray. 



145] A MEMOIR UPON CAUSTICS. 339 



11. 

But instead of investigating the nature of the caustic itself, we may begin by 
finding the secondary caustic or orthogonal trajectory of the refracted rays, i.e. a curve 
having the caustic for its evolute; suppose that the incident rays are all of them 
normal to a certain curve, and let Q be a point upon this curve, and considering 
the ray through the point Q, let G be the point of incidence upon the refracting 
curve ; then if the point G be made the centre of a circle the radius of which is 
fjT^ . GQ, the envelope of the circles will be the secondary caustic. It should be 
noticed that, if the incident rays proceed from a point, the most simple course is to 
take such point for the point Q. The remark, however, does not apply to the case 
where the incident rays are parallel ; the point Q must here be considered as the 
point in which the incident ray is intersected by some line at right angles to the 
rays, and there is not in general any one line which can be selected in preference 
to another. But if the refracting curve be a circle, then the line perpendicular to 
the incident rays may be taken to be a diameter of the circle. To translate the 
construction into analysis, let f, rj be the coordinates of the point Q, and a, 13 the 
coordinates of the point G, then f, 17, a, ^8 are in efifect functions of a single 
arbitrary parameter ; and if we write 

then the equation 

where x, y are to be considered as current coordinates, and which involves of course 
the arbitrary parameter, is the equation of the circle, and the envelope is obtained 
m the usual manner. This is the well-known theory of Gergonne and Quetelet. 



III. 

There is however a simpler construction of the secondary caustic in the case of 
the reflexion of rays proceeding from a point. Suppose, as before, that Q is the 
radiant point, and let G be the point of incidence. On the tangent at G to the 
reflecting curve, let fall a perpendicular from Q, and produce it to an equal distance 
on the other side of the tangent; then if q be the extremity of the line so produced, 
it is clear that g is a point on the reflected ray Gq, and it is easy to see that 
the locus of J is the secondary caustic. Produce now QG to a point Q' such that 
GQ' = QGy it is clear that the locus of Q' will be a curve similar to and similarly 
situated with and twice the magnitude of the reflecting curve, and that the two 
curves have the point Q for a centre of similitude. And the tangent at Q' passes 
through the point 5, Le. q is the foot of the perpendicular let fall from Q upon 
the tangent at Q\ we have therefore the theorem due to Dandelin, viz. 

43—2 



340 A MEMOIR UPON CAUSTICS. [l45 

If rays proceeding from a point Q are reflected at a curve, then the secondary 
caustic is the locus of the feet of the perpendiculars let fall from the point Q upon 
the tangents of a curve similar to and similarly situated with and twice the magni- 
tude of the reflecting curve, and such that the two curves have the point Q for a 
centre of similitude. 



IV. 

If rays proceeding from a point Q are reflected at a line, the reflected rays will 
proceed from a point q situate on the perpendicular let fall from Q, and at an equal 
distance on the other side of the reflecting line. The point q may be spoken of as 
the image of Q; it is clear that if Q be considered as a variable point, then the 
locus of the image q will be a curve equal and similar but oppositely situated to 
the curve, the locus of Q, and which may be spoken of as the image of such curve. 
Hence it at once follows, that if the incidental rays are tangent, or normal, or indeed 
in any other manner related to a curve, then the reflected rays will be tangent, or 
normal, or related in a corresponding manner to a curve the image of the first- 
mentioned curve. The theory of the combined reflexions and refractions of a pencil 
of rays transmitted through a plate or prism, is, by the property in question, rendered 
very simple. Suppose, for instance, that a pencil of rays is refracted at the first 
surface of a plate or prism, and after undergoing any number of internal reflexions, 
finally emerges after a second refraction at the first or second surface; in order to 
find the caustic enveloped by the rays after the second refraction, it is only necessary 
to form the successive images of the first caustic corresponding to the different reflexions, 
and finally to determine the caustic for refraction in the case where the incident 
rays are the tangents of the caustic which is the last of the series of images; the 
problem is not in effect different from that of finding the caustic for refraction in 
the case where the incident rays are the tangents to the caustic after the first re- 
fraction, but the line at which the second refraction takes place is arbitrarily situate 
with respect to the caustic. Thus e.g. suppose the incident rays proceed from a 
point, the caustic after the first refraction is, it will be shown in the sequel, the 
evolute of a conic; for the complete theory of the combined reflexions and refractions 
of the pencil by a plate or prism, it is only necessary to find the caustic by refraction, 
where the incident rays are the normals of a conic, and the refracting line is arbitrarily 
situate with respect to the conic. 



V. 

Suppose that rays proceeding from a point Q are refracted at a line; and take 
the refracting line for the axis of y, the axis of x passing through the radiant point 
Q, and take the distance QA for unity. Suppose that the index of refraction /a is 

put equal to r. Then if ^ be the angle of incidence and <(/ the angle of refraction. 



145] 



A MEMOIR UPON CAUSTICS. 



341 




we have sin <f>' = k sin <t>, and the equation y — x tan <f/ = tan <t> of the refiracted ray 
becomes, putting for ^' its value, 

y — 7-==== X — tan = 0. 
^ Vl-Jfc«sin«<^ ^ 

Differentiating with respect to the variable parameter and combining the two equations, 
we obtain, after a simple reduction, 



kx=-' 



{l-k' sin* <^)* 
cos*^ 

A'* sin' 



*'^""~ cos»0 ' 

where i' = Vl — A» ; hence eliminating, 

(Aa:)*-(A'y)*=l, 

which is the equation of the caustic. When the refraction takes place into a denser 
medium k is less than 1, and k'^ is positive, the caustic is therefore the evolute of 
a hyperbola (see fig. 1); but when the refraction takes place in a rarer medium k 
is greater than 1, and A'* is negative, the caustic is therefore the evolute of an 
ellipse (see fig. 2). These results appear to have been first obtained by Gergonne. 
The conic (hyperbola or ellipse) is the secondary caustic, and as such may be obtained 
as foUowa 



VI. 

The equation of the variable circle is 

aj" 4- (y - tan <^)» - A:" sec" = ; 
or reducing, the equation is 

^ ^y* - 2y tan <^ + A'« tan«<^- ifc« = : 

whence, considering tan as the variable parameter, the equation of the envelope is 

A:'«(aj« + y«-*»)-y» = 0, 
that is, 

jf'ar' - Jfcy - A^^jfc'" = 0, 



A HiaiOIR UPON CAUSTICS. 



is the equation of the secondary caustic, or conic haviog the caustic for its evol 
The radiant point, it is clear, is a focus of the conic 



VII. 
Let the equation of the refracted ray be represented by 
Xa:+Yy + Z = 0, 
ve have 



from which we obtain 



Vl - ft» sinV 



3> F'~2» 



for the tangential equation of the caustic ; or if we represent the equation of 
refracted ray by 

Zx+Yy-k = 0. 
then we have 

X' 7' f 
for the tangential equation of the caustic 



Fig. 1. 



Fig. 2. 




m 



A HBUOtR UPON CAUSTICS. 



vin. 



If a ray be reflected at a circle; we may take a, 6 as the coordinates of the 
centre of the circle, and supposing as before that f, 17 are the coordinates of a point 
Q in the incident ray, a, j8 the coordinates of the point of incidence, and x, y 
the coordinates of a point q in the reflected ray, the equation of the reflected ray, 
treating 3;, y as current coordinates, is 

((i-/3)(«-.)-(o-.)(y-»n(a-.)(f-.) + (i-«(l-S)l 

+ !(«-«)(«-.) +(i-«(y-(J))Ki-»(t-.)-(a-.)(,-^)).0. 
Write for shortness, 

*,..-('-»(«-«)-(»-a)(y-«. 

T,., .(a-i.)(»-a) + (i-S)(y-ffl, 
and similarly for Nf^^a &c. ; the equation of the reflected ray is 

Suppose that the reflected ray meets the circle agun in Q' and undergoes a 
second reflexion, and let jb', y* be the coordinates of a point q' in the ray thus twice 
reflected. We see first ((?' being a point in the first reflected ray) that 

Again, considering G as a point in the my by the reflexion of which the second 
reflected ray arises, the equation of the second reflected ray is 

uid from the form of the expressions Nq,a, '^q.a ^^ ^ clear that 

he equation for the second reflected ray may therefore be written under the form 

r reducing by a previous equation, we obtain finally for the equation of the second 
iflected ray, 

td in like manner the equation for the third reflected ray is 

1 n on, the equation for the last reflected ray containing, it will be observed, the 
tdinates of the radiant point and of the first and last points of incidence (the 
^dinates of the liist puitit of incidence can of course only be calculated &ora those 
: radiant point and thu first point of incidence, through the coordinates of the 
iate pqinU of incidence), but not containing explicitly the coordinates of any 
intermediate points of incidence. The form ia somewhat remarkable, but the 
I Is nal^ the same with that obtained by simple geometrical considerations, as 



^ 



344 



A MEMOIR UPON CAUSTICS. 



[145 



IX. 

Consider a ray reflected any number of times at a circle; and let G^G, be the 
ray incident at G, and GG' the last reflected ray, the point at which the reflexion 
takes place or last point of incidence being G. Take the centre of the circle for 




the origin, and any two lines Ox, Oy through the centre and at right angles to each 
other for axes, and let Ox meet the circle in the point A, Write 

/.AOG, =^0, Z.xG,G, = iro, 
ZAOO =0, ZxGG' =i/r, 
z GoG,0 = <l> ; 

then the radius of the circle being taken as the centre of the circle, the equation 
of the reflected ray is 

y — sin ^ = tan '^ (a? — cos 0) ; 

and if there have been n reflexions, then 

=^o + w(7r-2<^) = ^o +n7r-2n<^, 

•^ = '^o — 2n0, 

and therefore the equation of the reflected ray is 

y cos (iiro - 2n0) - x sin (i/to - 2n<^) + (-)** sin (-^o - ^o) = 0. 



X. 

If a pencil of parallel rays is reflected any number of times at a circle, then 
taking AO for the direction of the incident rays, we may write ^o = 0, '^o = 'n", and 
the equation of a reflected ray is 

X sin 2n<l> + y cos 2n^ = (—.)'* sin ^ ; 



145] A MEMOIR UPON CAUSTICS. 345 

differentiating with respect to the variable parameter, we find 

X cos 2n^ — y sin 2n^ = (— )** 5- cos ^ ; 
and these equations give 

X = ^^ I (2n + 1) cos (2n - 1) <^ - (2n - 1) cos (2w + 1) ^l , 

y = ^-^ I - (2n + 1) sin (2n - 1) + (2n - 1) sin (2w + 1 ) </>! , 

which may be taken for the equation of the caustic; the caustic is therefore an 
epicycloid: this is a well-known result. 



XL 

If ra3r8 proceeding from a point upon the circumference are reflected any number of 
times at a circle, then taking the point A for the radiant point, we have ao = 0, 
^^ = 'jr — <f>, and the equation of a reflected ray is 

a;sin(2n+ l)0-f y cos (2n + l)0 = (—)** sin <f>\ 

differentiating with respect to the variable parameter, we find 

^cos(2r{ + 1)0 — y sin(2n+ 1)0 = ( — )*»- -sin0; 

and these equations give 

(-)» ( ) 

a? = o '^ < (n + I)cos2n0— wco8(2n + 2)0- , 



(-Y ( ) 

y-^-Zi j-(w + l)8in2ii0 + ncos(2n + 2)0 



which may be taken as the equation of the caustic ; the caustic is therefore in this 
case also an epicycloid: this is a well-known result. 



XII. 

CJonsider a pencil of parallel rays refracted at a circle; take the radius of the 
circle as unity, and let the incident rays be parallel to the axis of x, then if 0, 0' 

be the angles of incidence and refraction, and /n or t be the index of refraction, so 

that sin 0^ = A; sin 0, the coordinates of the point of incidence are cos 0, sin 0, • and 
the equation of the refracted ray is 

y — sin = tan (0 — 0') (x — cos 0), 
le. 

cos (0 — 0O(y — sin 0) = sin (0 — if}') (x — cos 0), 
c. II. 44 



346 A MEMOIR UPON CAUSTICS. [145 

or 

y cos {<f> — 0') — a? sin (^ — if>) = sin 0', 

which may also be written 

(y cos <^ — a? sin 0) cos 0' + (y sin + a? cos — 1) sin ^' = 0. 



Writing k sin ^, vT— A;* sin' j> instead of sin if/, cos <f>\ and putting for shortness 

y cos — a? sin = F, 
y sin + a? cos <^ = X, 

A? sin _ . 
Vl - ^> sin* <^ 

the equation of the refracted ray becomes 

F+4>(Z-1) = 0; 

and differentiating with respect to the variable parameter 0, observing that 

dF_ y dX _ «. 

d<^ ' d(f> 

d4>_ icos<<^ _ cot^ . 
# " (1 - A:> sin«^)* "" 1^ A:* sin* if> ' 



we have 



V 1 - if 81U' / ' 



4> 

and the combination of the two equations gives 



y.^_^(lj-A^8in»^ 
<Pcot</>-l ' 

^ _ 4> c ot (^ — A;* si n' 
4>cot<^-l " • 



and we have therefore 



TT ^ , v • ^ Ar'sm'^^^coti^-l) , . . , . 

V=xcos6 + XsinA= ^\ , — r ^ = A:^ sin* 6, 

^ ^ ^ 4>cot<^ — 1 ^ 

* ( -. — 7 - A;* sin* 6 I - A;* sin* 6 cos 6 

a? = Xcos6— Fsm A= ^^- -g- ; — ; 

^ ^ 4> cot <^ - 1 

i.e. 

_ ^ (1 — A;* sin* <b) — A:* sin* <^ cos ^ 
~ <P cos — sin ' 



145] A MEMOIR UPON CAUSTICS. 347 

or multiplying the numerator and denominator by (1 — A:* sin* <^) (4> cos ^ + sin <^), the 
numerator becomes 

(1 - A:* sin* <^) {** cos <^ (1 - A;* sin* <^) - A:* sin* <^ cos if} 

+ * (sin <^ (1 - A;* sin* 0) - A:* sin* ^ cos 0)} 
= A:* sin* </> cos {(1 - A;* sin* 0) - sin* (1 - A;* sin* 0)} 

+ k sin* Vl - A:* sin* ^ (1 - A;* an* 0) 
= A* sin* cos* + A? sin* ^ (1 - A:* sin* 0)*, 
and the denominator becomes 

A;* sin* ^ cos* <^ — (1 — A;* sin* 0) sin* <f) 
= - A/* sin* </>, 
if A/* = l-A:*. 

Hence we have for the coordinates of the point of the caustic, 

[A/«a? = - A:* cos* <^ - A; (1 - A?* sin* <^)*, 
y= A?* sin* ^ ; 

and eliminating 0, we obtain for the equation of the caustic, 

A;'«ar = -A^{l-.A;-*y*}*-A;{l-A;V}*; 
or writing - instead of k, we find 

(i-/.*)^=(i-/iV)*+/^(i-/*"*y¥ 

for the equation of the caustic by refraction of the circle, for parallel raya The 
equation was first obtained by St Laurent. 



XIII. 

The discussion of the preceding equation presents considerable interest. In the 
first place to obtain the rational form write 

a =(i-/iO^, )8 = (i-/.V)^7=/^(i-/*-*y¥; 

this gives 

a* - 20? (/S" + 7*) + (/S* - 7')' = 0, 
and we have 

/3« = 1 - 3/i V + 3/tV - yity, 

rf ^fi*- 3/t%* + 3/t V - y», 
and consequently 

/3« - y = (I - /I') {1 - 3/iV + (!+/*') y'i- 

44—2 



348 A MEMOIR UPON CAUSTICS. [l45 

Hence dividing out by the factor (1 — fi^y, the equation becomes 

(l-/Lt«)«a;*-2(l+;i«-6AtV+3/^*(l+/**)y*-(l+/^')j^)2a?" + (l-3AtV + ^^ 
or reducing and arranging, 

+ (12/tV-h9AtV)y^-(6/^*(l+/^*)a^ + 6At* + 6At*(l+At»)y»)y* = 0, 
which is of the form 

A + 3/Lt*5y* - QfjfiCy* = ; 
and the rationalized equation is 

4* + 27 fi*B'i/' - 21 6/i«(?y* + Bi^fi^ABCy* = 0, 
where the values of A, B, C may be written 

4 = (^ + y»){(l-/.«)»^ + (H-/i»)»y»}-2(l+/.»)(a;»-y») + l, 

5 = 4«r« + 3y«, 

the caustic is therefore a curve of the 12th order. 

To find where the axis of x meets the curve, we have 

y=0, 4o' = 0, 
where 

= {(!-,.)» a;--l}{(l+/i)»^-l}, 
i.e. 



fy-O. ^ 



X— ± = , ir= + 



1 " fi' 1 4-/a' 

or there are in all four points, each of them a point of triple intersection. 

To find where the line oo meets the curve, we have 

where 

i.e. 

^ 1 + /i» . 



145] A MEMOIR UPON CAUSTICS. 349 

or the curve meets the line oo in four points, each of them a point of triple 
intersection: two of these points are the circular points at oo. 

To find where the circle as* + y* = 1 meets the curve, this gives a;* = 1 — y*, and 
thence 

5 = 4-y», 

and the equation becomes 

{/A« (/*' - 4) + 4 (1 + 2/i«) y«}« + 27/A* (4 - y'/ y» - 216 (/A« + 2)» y* 

+ 54At»(Ai« + 2)y»(4-y»){/Lt«(Ai«-4) + 4(l+2At»)y»}=0, 

which is only of the eighth order; it follows that each of the circular points at oo 

(which have been already shown to be points upon the curve) are quadruple points 

of intersection of the curve and circle. The equation of the eighth order reduces 
itself to 

(y»-/iy{27Aiy + (M'-4)»}=0; 

the values of x corresponding to the roots y = + /a are obtained without difficulty, 
and those corresponding to the other roots are at once found by means of the 
identical equation 

(/Lt«-4)» + 27/i* + (l-/i»)(^« + 8)» = 0; 

we thus obtain for the coordinates of the points of intersection of the curve with 
the circle a;* + y' = l, the values 



a? = ± Vl - ^» 



foo, |a?=±V: 

(a?=±ty, (y=±/f, 



x= ± = t, 

^ (m' - 4)* . 



each of the points of the first system being a quadruple point of intersection, each 
of the points of the second system a triple point of intersection, and ectch of the 
pomts of the third system a single point of intersection. 

1 1 

Next, to find where the circle a;*4-y*=r— meets the curve; writing a;* = — — y», 

I* fit 

we obtain for y an equation of the eighth order, which after all reductions is 

(y* - ^)* {27/ty + (1 - v/} = 0, 



350 



A MEMOIR UPON CAUSTICS. 



[145 



and we have for the coordinates of the points of intersection, 



( 



{:= 



00 

a? = ± iy, 



X 






1 



(1 + 8m') 



«= + 



y = 






^ (1 - V)* • 



?= ». 



each of the points of the first system being a quadruple point of intersection, each 
of the points of the second system a triple point of intersection, and each of the 
points of the third systemi a single point of intersection. 

The points of intersection with the axes of a?, and the points of triple inter- 
section with the circles aj" + y'=l and a;*-fy« = — , are all of them cuspidal points; 

the two circular points at oo are, I think, triple points, and the other two points of 
intersection with the line oo, cuspidal points, but I have not verified this: assuming 
that it is so,' there will be a reduction 54 accounted for in the class of the curve, 
but the curve is, in fact, as will be shown in the sequel, of the class 6 ; there is 
consequently a reduction 72 to be accounted for by other singularities of the curve. 



XIV. 

It is obvious from the preceding formulae that the caustic stands to the circle^^ 
radius -, in a relation similar to that in which it stands to the circle, radius 1, Le. 
to the refiracting circle. In fact, the very same caustic would have been obtained if^^ 
the circle radius — had been taken for the refracting circle, the index of refraction 

being ~ instead of ^. This may be shown very simply by means of the irrational 
form of the equation as follows. 

The equation of the caustic by refraction of the circle, radius 1, index of refraction 
/ir, is as we have seen 

(i-/.>)fl:=(i-/iW+M(i-/*-»y¥; 

hence the equation of the caustic by refiuction of the circle radius o\ index of 
refiwjtion /i', is 

(.-.^,i.{.-.-.(i)'}'...{i-.-.(|)y, 



145] 



A MEMOIB UPON CAUSTICS. 



351 



or, what is the same thing, 

which becomes identical with the equation of the first-mentioned caustic if /a' = c' = — . 

Hence taking c instead of 1 as the radius of the first circle, we find, 

Theorem. The caustic by refi-action for parallel ra}rs of a circle, radius c, index 
of refraction /i, is the same curve as the caustic by refraction for parallel rays of a 

concentric circle, radius - , index of refraction - . 



XV. 

We may consequently in tracing the caustic confine our attention to the case in 

which the index of refiuction is greater than imity. The circle, radius -, will in this 

case be within the refracting circle, and it is easy to see that if from the extremity 
of the diameter of the refiucting circle perpendicular to the direction of the incident 

rays, tangents are drawn to the circle, radius - , the points of contact are the points 

of triple intersection of the caustic with the last-mentioned circle, and these point-s 
of intersection being, as already observed, cusps, the tangents in question are the 
tangents to the caustic at these cusps. The points of intersection with the axis of 
X are also cusps of the caustic, the tangents at these cusps coinciding with the axis 
of x: two of the last-mentioned cusps, viz. those whose distances from the centre are 

1 . . . . . c 

± , lie within the circle, radius - , the other two of the same four cusps, viz. 

M + 1 M 



those whose distances from the centre are ± 



M-l 



, lie without the circle, radius 



- ; the last-mentioned two cusps lie without the refracting circle, when /a < 2, upon 

this circle, when /a = 2, and within it and therefore between the two circles, when 
/A>2. The caustic is therefore of the forms in the annexed figures 3, 4, 5, in each 



Fig. 3. 



Fig. 4. 



Fig. 5. 



T"-^--. 




**.. 





352 A MEMOm UPON CAUSTICS. [145 

of which the outer circle is the refracting circle, and fi ib > 1, but the three figures 
correspond respectively to the cases ft < 2, ft = 2 and /k > 2. The same three figures 
will represent the different forms of the caustic when the inner circle is the refracting- 
circle and fi is < 1, the three figures then respectively corresponding to the cases 

A^ > i> /^ = i> ^^ /* < i- 



XVI. 

To find the tangential equation, I retain k instead of its value -; the equation 
of the refructed ray then is 

a? (A; cos - Vl - ^in« <^) + y (k sin <^ + cot ^ Vl - A:» sin* <t>) - Jfc = 0, 

and representing this by 

Zar+Fy-A;=0, 
we have 

Z = A;cos0-Vl-A;»sin»<^, 

F = A: sin ^ 4- cot ^ Vl — A:* sin* 0, 



equations which give 



X cos + Fsin ^ = A?, 

z«+r* ^ 



and consequently 



sin* <l> ' 



. . 1 

sm = ,— 

, Vz*+y*-i 

cos 6 = — .— — - 
^ VZ* + F* 



^ ) 



and we have 



which gives 



ZVZ*+ F*-l + F-ifc VZ*+ F* = 0, 



(Z«+F*)(Z*-l-Jfc*) = -2A;FVZ*+F*; 
or, dividing out by the factor VZ* + F*, the equation becomes 

\/Z*+F*(Z*- 1-Jfc*) = - 2A;F, 
from which 

(Z*+F*)(Z*-l-Jfc*)*-4ifc»F* = 0; 

or reducing and arranging, we obtain 

Z*(Z*-l-Aj*)*+F*(Z + l+A;)(Z + l-A:)(Z-l+ifc)(Z-l-i) = 

for the tangential equation of the caustic by refraction of a circle for parallel raj^s. 
The caustic is therefore of the class 6. 



145] A MEMOIR UPON CAUSTICS. 353 



XVII. 
Suppose next that rays proceeding from a point are reflected at a circle. 

A very elegant solution of the problem is given by Lagrange in the Mim. de 
Turin; the investigation, as given by Mr P. Smith in a note in the Cambridge and 
Dublin Mathematical Journal, t. ii [1847] p. 237, is as follows: 

Let B be the radiant point, RBP an incident ray, and PS a reflected ray; CA 
a fixed radius; ACP = a, ACB = €, reciprocal of CB = Cy reciprocal of CP = a, The 

equations of the incident and reflected ray, where w = - , may be written 




u=^Asin0 + B cos ; incident ray, 

t* = i4sin(2a — ^) + 5cos(2a — ^); reflected ray, 

the conditions for determining A and B being 

a = ^ sin a 4- 5 cos a, 

c = -A sin e + -B cos e, 

whence 

. _ a cos € - c cos a p _ c sin a — a si n e 
sm (a — e) sin (a — e) 

Substituting these values, the equation of the reflected ray becomes 

a sin (2a — ^ — e) = M sin (a — 6) 4- c sin (a — ^), 

firom which and its differential with respect to the arbitrary parameter ce, the equation 
of the caustic, or envelope of the reflected rays, will be found by eliminating a. 

In this, a being the only quantity treated as variable in the differentiation, let 

2a-^-e = 20, 
therefore 



a iL 



45 



354 A MEMOIR UPON CAUSTICS. [l45 

and the equation becomes 

a sin 20 = w sin {<^ + J (^ - €)} + c sin [<f>-^{0- e)}. 



Make 



also 



p __ (<f + c) cos ii^ — e) 
^ 2a 



] 1 



then the equation becomes 



with the condition 



Hence 



cos ' ^ sin (^ ' 



P^ + Qy = i. 



•»"'+y"*=i. 



multiplying by x and y, and adding, we find X = 1 ; therefore 

Hence 

or restoring the values of P and Q, 

{(16 + c) cos i (^ - e)}* + {(u - c) sin ^{0- e)}* = 1, 
the equation of the caustic. 



XVIII. 

But the equation of the caustic for rays proceeding from a point and reflected 
at a circle may be obtained by a different method, as follows: 

Take the centre of the circle for origin ; let c be the radius of the circle, fl, ^ 
the coordinates of the radiant point, a, the coordinates of the point of incidence, 
X, y the coordinates of a point in the reflected ray. Then we have from the equation 
of the circle o^-\-^'=^&, and the equation of the reflected ray is by the general 
formula, 

(ft'x - a/8) (cm? + /8y - c») + (ya - a?/8) (aa + 6/8 - c*) = ; 



145] A MEMOIR UPON CAUSTICS. 355 

or arraDging the terms in a diflferent order, 

(6a? + ay)(a»-)8') + 2(6y-cur)a^-c^(6+y)a + c»(a + a?))8 = 0; 

and writing herein a = c cos 0, /8 = c sin ^, the equation becomes 

(bx + ay) cos 2^ + {hy — olx) sin 2^ — (6 + y) c cos ^ + (a + a:) c sin ^ = 0, 
where d is a variable parameter. 

Now in general to find the envelope of 

-A cos 2^ + J5 sin 2^+ C cos ^ + 2) sin ^ + ^ = 0, 
we may put e^==z, which gives the equation 

and equate the discriminant to zero : this gives 

(4/)» - 27 (- 8J)» = 0, 
where 

-8J = 4((?-2>»)+25C'2)--{8(^'' + £«) + ((? + 2>»)}i^+^^, 
and consequently 

- 27 [A ((?- Z)«) + 2BGD - (8 {A^^B") + (0+ Z>«)) ^^ + ^J5?}« = 0; 

«nd substituting for Ay B, C, D, E their values, we find 

f 4 (a« + 6») (aj" + y*) - c« ((a? + a)« + (y + 6)»)}'- 27 (bx-ayy (a^ + f- a^^¥y=- 0, 

ibr the equation of the caustic in the case of rays proceeding from a point and 
xeflected at a circle: the equation was first obtained by St Laurent. 

It will be convenient to consider the axis of ^ as passing through the radiant point ; 
this gives 6 = 0; and if we assume also c = l, the equation of the caustic becomes 

{(4a« - 1) (aj" + y«) - 2gw? - a«}» - 27ay (ic» + y»- a»)» = 0. 



XIX. 

Reverting to the equation of the reflected ray, and putting, as before, c = 1, 6 = 0, 
tihis becomes 

/ « /I . 1 \ a cos 2^ — cos ^ ^ 

(-2acos^ + l)a?+ ^— ^ t/ + a = 0; 

^ ^ sm r "^ 

> 

differentiating with respect to 0, we have 



(-2asm^)ar + . ,^ y = 0; 



45—2 



356 



and from these equations 



A MEMOIR UPON CAUSTICS. 



[■ 



_ tt^cos (1 -h28 i n«g)~a 
^ " 1 -~3a cos 2dT2a« 



y = 



2a« sin» ^ 



1- 3a cos 2^+ 2a*' 



which give the coordinates of a point of the caustic in terms of the angle wl 
determines the position of the point of incidence. The values in question satisfy, 
they should do, the equation 

{(4a» - 1) (aj" + y«) - 2aa? - a»}» - 27ay (ic» + y» - a«)« = 0; 

we have, in fact, 

4a* (cos 0—af 



^+y' 



-tt» = 



(4a«- 1) (ar» + y«) - 2aa; - a' = 



(1 - 3a cos 2^ + 2a«)» ' 
12aHcos^-ay 



(l-3acos2^+2a«)>' 
from which it is easy to derive the equation in question. 



XX. 

If we represent the equation of the reflected ray by 

Xx-^Yy + a- 0, 



then we have 



X = -2acos^ + l, 
a cos 20 — cos 



F= 



sin 



and thence 



(Z - 1)> - 4a« = - 4a« sin« ^, 

X'+Y* = -.-\ >, (1 - 2a cos + a»), 

sm^ 0^ 

X + a^ =l-2aco80 + a\ 

and consequently 

(X*+Y') {(Z- 1)»- 4a»l + 4a»X + 4a* = 0, 

or, what is the same thing, 

{Z(Z-l)-2a«}«+F='{(Z-l)«-4a'} = 0, 

which may be considered as the tangential equation of the caustic by refle: 
circle; or if we consider X, F as the coordinates of a point, then the equ; 
be considered as that of the polar of the caustic. The polar is therefore a cv 
fourth order, having two double points defined by the equations X(Z — 1) — 2a* 



145] A MEMOIR UPON CAUSTICS. 357 

and a third double point at infinity on the axis of F, i.e. three double points in all ; 
the number of cusps is therefore 0, and there are consequently 4 double tangents and 
6 inflections, and the curve is of the class 6. And as F is given as an explicit 
function of X, there is of course no difficulty in tracing the curve. We thus see 
that the caustic by reflexion of a circle is a curve of the order 6, and has 4 double 
points and 6 cusps (the circular points at infinity are each of them a cusp, so that 
the number of cusps at a finite distance is 4) : this coincides with the conclusions 
which will be presently obtained by considering the equation of the caustic. 



XXI. 

The equation of the caustic by reflexion of a circle ia 

{(4a« - 1) (a:» + y«) - 2cw? - a»}» - 27ay (ar» + y* - a^y = 0. 

Suppose first that y = 0, we have 

{(4a« - 1) a:* - 2cw? - a*}« = 0, 
— a __ a 

^^' ^'isr+i' ^"2^-r 

or the curve meets the axis of a? in two points, each of which is a triple point of 
intersection. 

Write next aJ" + y'=a^ this gives 

{(4a« - 1) a« - 2aa? - a«}» = 0, 

and consequently 

a? = - a (1 - 2a«), 

y = ± 2a» Vi - a«, 

or the curve meets the circle a;* + y*— a^ = in two points, each of which is a triple 
point of intersection. 

To find the nature of the infinite branches, we may write, retaining only the terms 
of the degrees six and five, 

(4a«- l)'(a;» + y»)»- 6 (4a« - l)«a (a;» + y*)» a?- 27 ay (a;» + y*)^ = ; 
and rejecting the factor (oc^ + jf^y, this gives 

(4a»-l)»«»+{(4a«-l)»-27a«}y»-6(4a»-l)»aa: = 0; 
or reducing, 

(4a«-l)»a;»-(l-a')(8a» + l)»y»-6(4a*-l)»aa?=0; 

and it follows that there are two asymptotes, the equations of which are 

_(4^-_l)*_ L 3^ 1 
VT^*(8a' + l)l 4a«-ir 



358 A MEMOIR UPON CAUSTICS. [l45 

Represent for a moment the equation of one of the asymptotes by y = A{x — o), 

then the perpendicular from the origin or centre of the reflecting circle is iia-s- Vl -\-A^, 
and 

3a V4a* - 1 



Aa = 



Vl-a«(l+8aO 



I , ^,^ (l-a')(l+Sa«)' + (4a«-l) »^ 27a' 



Vl ^A^ = 



(1 - a«) (1 + 8a»)« (1 - a«) (1 + 8a»)^ ' 

3V3a 



Vl - a« (1 + 8a«) 



and the perpendicular is ^V4r^»_i, which is less than a if only a«<l, i.e. in every 
case in which the asymptote is real. 

The tangents parallel and perpendicular to the axis of x are most readily obtained 
from the equation of the reflected ray, viz. 

/ « /» 1 \ a cos 2^ — cos ^ ^ 

{— 2a cos ^+1) a; H : — -^ v + a = 0: 

the coefficient of x (if the equation is first multiplied by sin^) vanishes if sin = 0, 

1 V4a' — 1 
which gives the axis of a?, or if cos = — , which gives y = ± — ^ , for the tangents 

parallel to the axis of x. 

The coefficient of y vanishes if a cos 20 — cos 5 = 0; this gives 

^ 1 + V8a« + 1 . ^ 1 ,- , , ^ /s-5 — ?x 

cosg= - ^ , 8ine=^- (4a»-l T v8a«+l), 

4a oar 

and the tangents perpendicular to the axis of x are thus given by 

-2a 



1 + V8a« + 1 ' 

these tangents are in fact double tangents of the caustia In order that the point of 
contact may be real, it is necessary that sin 0, cos should be real ; this will be the 
case for both values of the ambiguous sign if o > or = 1, but only for the upper 
value if a < 1. 

It has just been shown that for the tangents parallel to the axis of x, we have 

V4a«-1 



y = ± 



2a 



V4a' — 1 
the values of y being real for a > ^ : it may be noticed that the value y = — 



145] A MEMOIR UPON CAUSTICS. 359 

is greater, equal, or less than or to y = 2a' Vl — a', according as a > = or < 7= 5 ^^^ 
depends on the identity (4a» - 1) - 16a« (1 - a«) = (2a» - 1)« (2a» +1). 

To find the points of intersection with the reflecting circle, ic* + y' — 1 = 0, we have 

(3a« - 1 - 2cw:)»- 27a» (1 - a;»)(l - a«>» = ; 
or, reducing, 

8aV + (- 27a* + 18a»- 15) a'ic* + (54a* - 36a» + 6) oo? + (- 27a* + 18a«+ 1) = 0, 
Le. (aa?-l)»(8aa?-27a*+18a» + l) = 0. 

The factor (ax^iy equated to zero shows that the caustic touches the circle in 

the points a? = - , y = ± a/ 1 ^ , i e. in the points in which the circle is met by the 

polar of the radiant point, and which are real or ima^nary according as a > or < 1. 
The other &ctor gives 

27a* - 18a« - 1 



x = 



8a 



Putting this value equal to ± 1, the resulting equation is (a + l)(27a' + 9a + l) = 0, and 
it follows that x will be in absolute magnitude greater or less than 1, Le. the points 
in question will be imaginary or real, according as a>l or a<l. 

It is easy to see that the curve passes through the circular points at infinity, 
and that these points are cusps on the curve ; the two points of intersection with the 
axis of X are cusps (the axis of x being the tangent), and the two points of inter- 
section with the circle a^ + y' — a' = are also cusps, the tangent at each of the cusps 
coinciding with the tangent of the circle ; there are consequently in all 6 cusps. 



XXII. 

To investigate the position of the double points we may proceed as follows: write 
for shortness P = (4a*-l)(a:* + y')- 2aa?-a*, Q = ar/S, S^-x-k-f-a}) the equation of the 
caustic is 

P»-27Q» = 0; 
hence, at a double point, 

one of which equations may be replaced by 

dP dQ^dP dQ^^ 
dx dy dy dx ' 



360 A MEMOIR UPON CAUSTICS. [145 

Now — 

^£ = 2{(^'-l)x-a], ^| = 2(4a'-l)y. 

^=2aay. ^ = o(^ + 3y'-a») = a(S+2y«); 

substituting these values in the last preceding equation, we find 

~(4a»-l)"y "S-h2y«* 
or, reducing, 

and using this to simplify the equation 



we have 



^s-'««S-». 



P'^-lSay8.2axy = 0, 



i.e. ^ -^ — 9cuF/S> = 0, 

and therefore 

Multiplying by P and writing for P* its value 27a^y'S^, we have 

Px = 3ay«, 
and thence 



whence 






«y* = 5» ^ = S or 2^=?!- 

a!» • J • S „i' 



or "a' 

and substituting in the equation 

* 4a'-U Sj' 
we find 

or, rationalising, 

4(M!» - {(4o' - 1) « - o)' = 0, 



145] A MEMOIR UPON CAUSTICS. 361 

or, what is the same thing, 

(4flw?-l)(a?-a(a + iVl -a«)») (a:-a(a-iVl -a»)«) = 0. 

The factor 4tax — l equated to zero gives ^=7- from which y may be found, but the 

resulting point is not a double point; the other factors give each of them double 
points, and if we write 

we find 

^ 2aH'(a + iVl-a')* 

values which, in fact, belong to one of the four double points. It is easy to see that 
the points in question are always imaginary. 

It may be noticed, by way of verification, that the preceding values of x, y give 
(4a« - l)(«" + y') - 2flw: - a» = ^^^^^ (1 - 4a»- 4ai VTIT^), 

x' + y»-a' = j-j-g^, (3a + i Vl - a« ), 

f = ^t."^, (- 1 + 14a» - 16a« + 2a (3 - 8a») i Vl - a*) ; 

and if the quantities within ( ) on the right-hand side are represented by A, B, C, then 

i (a + iVrr^'). 



whence we have identically, 



B 



f) 

_ = _(a4-zVl-a»)>, 



(^)' = 5' - ^'=^^' 



by means of which it appears that the values of x, y satisfy, as they should do, the 
equation of the caustic ; and by forming the expressions for (4a'— l)a; — a and a^ + S^*— a', 
it might be shown, d posteriori^ that the point in question was a double point. 

XXIII. 
The equation 

{(4a«- 1) (a;* + y«) - 2aa? - a«}» - 27ay (a^ + y' - a«)» = 

l)ecome8 when a = 1 (L e. when the radiant point is in the circumference), 

{3y> + (a? - 1) (3a?+ 1)}» - 27y« (y» + «» - 1)» = ; 

it is easy to see that this divides by (a?— 1)*; and throwing out this factor, we have 
for the caustic the equation of the fourth order, 

27 j^ + 18y« (3ar» _ 1) + (a? - 1) (3a; + 1)» = 0. 



A /• 



362 



A MEMOIR UPON CAUSTICS. 



[145 



XXIV. 

The equation 

{(4a« - 1) (a^ + jf") - 200? - a»}» - 27ay (ic* + y» - a«)» = 

becomes when a = x (i. e. in the case of parallel rays), 

(4ic» + 4y«-l)»-27y» = 0. 
which may also be written 

64>af + 48a?* (4y« - 1) + 12a?»(4y» - 1)» + (8y« + 1)» (y* - 1) = 0. 



XXV. 

It is now easy to trace the curve. Beginning with the case a = oo , the curve lies 
wholly within the reflecting circle, which it touches at two points; the line joining 
the points of contact, being in fact the axis of y, divides the curve into two equal 
portions ; the curve has in the present, as in every other case (except one limiting 
case), two cusps on the axis of x (see fig. 6). Next, if a be positive and > 1, the 
general form of the curve is the same as before, only the line joining the points of 
contact with the reflecting circle divides the curve into unequal portions, that in the 

Fig. 7. a>l. 



Fig. 


6. 


a — CD. 


,• 


i^S^^ 


1 ^^^."^^ 


yy 






/ / 




j \\ 


/ V 




[ J \ 


i f 




. .-..^....f. 


\ 1 




1 \ ' 


\ I 




I J / 


\ \ 




• / / 


*» V 




1 y ^ 


''C^ 


Sw 






neighbourhood of the radiant point being the smaller of the two portions (see fig. 7). 
When a = 1, the two points of contact with the reflecting circle unite together at the 
radiant point ; the curve throws off, as it were, the two coincident lines a? = 1, and the 
order is reduced fi-om 6 to 4. The curve has the form fig. 8, with only a single cusp 

on the axis of x. If a be further diminished, a < 1 > -= , the curve takes the form 

v2 

shown by fig. 9, with two infinite branches, one of them having simply a cusp on 
the axis of x, the other having a cusp on the axis of m^ and a pair of cusps at its 
intersection with the circle through the radiant point; there are two asymptotes equally 

inclined to the axis of x. In the case a — -j=^^ the form of the curve is nearly the 

same as before, only the cusps upon the circle through the radiant point lie on the 

axis of y (see fig. 10). The case a<-~>\ is shown, fig, 11. For = ^, the two 



145] 



A MEMOIR UPON CAUSTICS. 



363 



asymptotes coincide with the axis of a?; one of the branches of the curve has wholly 
disappeared, and the form of the other is modified by the coincidence of the asymptotes 



Fig. 8. a=l. 





Fig. 13. a = l 




\ 

\ \ 



Fig. 11, 



^ 1 




-v A- 



.».,—>.. 



i X-4- 




Fig. 10. a 



V2' 



Fig. 12. a = J. 





\ 



•- — i — 



with the axis oi x\ it has in fact acquired a cusp at infinity on the axis of x (see 
fig. 12). When a<\, the curve consists of a single finite branch, with two cusps on 
the axis of x^ and two cusps at the points of intersection with the circle through 
the radiant point; one of the last-mentioned cusps will be outside the reflecting circle 
as long as o>J; fig. 13 represents the case a=^, for which this cusp is upon the 
reflecting circle. For a<J, the curve lies wholly within the reflecting circle, one of 
the cusps upon the axis of x being always within, and the other always without the 
circle through the radiant point, aod as a approaches the curve becomes smaller 
and smaller, and ultimately disappears in a point. The case a negative is obviously 
included in the preceding one. 

Several of the preceding results relating to the caustic by reflexion of a circle 
were obtained, and the curve is traced in a memoir by the Rev. Hamnet Holditch, 
(^Tierly MathemcOical Journal, t i. [1857, pp. 93 — 111]. 

46—2 



364 



A MEMOIR UPON CAUSTICS. 



[145 



XXVI. 

Suppose next that rays proceeding from a point are refraxiied at a circle. Talie 
the centre of the circle as origin, let the radius be c, and take ^, i; as the coordi- 
nates of the radiant point, a, /9 the coordinates of the point of incidence, x, y the 
coordinates of a point in the refracted ray : then the general equation 



-qG VQON +n*QO '7qON =0 

becomes, taking the centre of the circle as the point N on the normal, or writinj^ 
a=b, 6 = 0, 

or putting a*-(-y3' = C, and expanding, 

a' {2 (i,»ar - /iVf )} 
+ a«/8 {- 4 (^x - /i'an/f ) + 2(ryy- ^*fn)\ 
+ a/S* {- 4 (fi,y - ii}xyf,) + 2 (pa: - /iVf)} 

-o« {(a!» + y» + c»)i?» -M* (?+'?* + C)!^} 
+ 2a^8 ((a^ + y» + c») fv - /*« (f» H- 1;» + c*) ay} 
-/3» {(«»-l-y' + c»)p-/t«(f» + i7» + c»)a?} 
= 0, 
which may be represented by 

4a« + P««/8-|- Ca/g' + D/S'-l- .P«» + ^0/9 + H0' = O. 
Now a* + /8* = c*, and we may write 

The equation thus becomes 

-l<-J)'-!«(-l)('-i)-H'-i)'-» 



or expanding, 



+ -(F-Gi-H) ^ 

+ (3^-£t + C + 3i)i) « 
+ *(F+H) 



^=0. 



+ (3il + £» + C - 3IH") - 



-l--(.P-(-(?i-5) 
c 



1 



+ (4 + 5i - C + IH) i 



145] A MEMOIR UPON CAUSTICS. 365 

n which z may be considered as the variable parameter ; hence the equation of the 
austic may be obtained by equating to zero the discriminant of the above function 
>f z\ but the discriminant of a sextic function has not yet been calculated. The 
equation would be of the order 20, and it appears from the result previously obtained 
or parallel rays, that the equation must be of the order 12 at the least ; it is, I think, 
)robable that there is not any reduction of order in the general case. It is however 
)racticable, as will presently be seen, to obtain the tangential equation of the caustic 
)y refraction, and the curve is thus shown to be only of the class 6. 



XXVII. 

Suppose that rays proceeding from a point are refracted at a circle, and let it be 
•equired to find the equation of the secondary caustic: take the centre of the circle as 
>rigin, let c be the radius, f , i; the coordinates of the radiant point, a, fi the coordinates 
)f a point upon the circle, /la the index of refraction; the secondary caustic will be 
ihe envelope of the circle, 

where a, /8 are variable parameters connected by the equation a' + zS' — c* = 0; the 
liquation of the circle may be written in the form 

But in general the envelope of -4a + £/8 + C = 0, where a, /8 are connected by the 
equation a' + ^S* — c'ssO, is & {A} + JP) — (? = 0, and hence in the present case the equation 
of the envelope is 

which may also be written 

{/it« (a^ + y» - c*) - (p + 17» - c«)}« = 4c«/it« {(a; - f )« + (y - 17)»}. 

If the axis of x be taken through the radiant point, then 97 =0, and writing also 
{ = a, the equation becomes 

or taking the square root of each side, 

{/A« (a;* + y» - c») - a« + c"} = 2cm V(a?-a)» + y« ; 

1 . / IV 

whence multiplying by 1 — 5 and adding on each side c* ( /^ — ) + (^ — a)^ + y', we have 

/**|(«-^,y + yj = |V(^-a)« + y« + c(M-J)}', 



or 



'^V (^~5) +y'='^(^-«)"+y'+^(^-^)' 



which shows that the secondary caustic is the Oval of Descartes, or as it will be con- 
venient to call it, the Cartesian. 



366 A MEMOIB UPON CAUSTICS. [145 

It is proper to remark, that the Cartesian consists in general of two ovals, one 
of which is the orthogonal trajectory of the refracted rays, the other the orthogonal 
trajectory of the false refracted raya In the case of reflexion, the secondary caustic 
is a Cartesian having a double point ; this may be either a conjugate point, or a real 
double point arising from the union and intersection of the two ovals ; the same 
secondary caustic may arise also from refraction, as will be presently shown. 



XXVIII. 

Reverting to the original form of the equation of the secondary caustic, multipljring 

by — J (1 i] and adding on each side ~fl ij +~, {(^ — a)* + y'j, the equation 

becomes 

or extracting the square root, 

Combining this with the former result, we see that the equation may be expressed 
indifferently in any one of the four forms, 



^/(-f)'-^'=^;:^/(^-J)>^'-^- 






(" - ^) \/(' - 9" - '• ^ (- <■ + ^ \/(' - J.)' - ^ - e - t) ^'<^^^>^ = «■ 



It follows, that if we write successively 

a'= a, c' = c f /*' = /* (1) 

a = - , c = - , M = - (a) 

a fi a ^ ^ 

a = - 2 , c = , /A = - (p) 

a =a , c = - , M = - (7) 

/x c 

a=— , C =C y a = — (0) 



145] A MEMOIR UPON CAUSTICS. 367 

or what is the same thing, 

(1) 

(a) 

(0) 

(7) 



or what is again the same thing, 



a==a' , 


c = c', 


t*=l*' 


a' 


a' 

c= -, 

A* 


a' 


of 


c' 


1 


a =a' , 


a' 


a' 


a =->, 
a 


c = c' , 


ey 

a 


a 


c' 


c' 
a 


J thing, 






o' = a. 




a' a 


a 


c'» a 
a' /*»' 


a' 




a a 


a' 
;.'. = « 


a' = a , 


c'' a 




a 


a 


a' a 




a 


/» a 



(8) 



(«). 



(1) 
(«) 
(/8) 

(7) 

(S) 

(0. 

we have in each case identically the same secondary caustic, and therefore also 
identically the same caustic ; in other words, the same caustic is produced by six 
different systems of a radiant point and refracting circle. It is proper to remark that if 
we represent the six systems of equations by (a', c', fi!) = (a, c, /i), {a\ c', /i') = a (a, c, /a), 
&c., then, a, )3, 7, S, 6 will be functional symbols satisfying the conditions 

a = /S" = &y = eS =76, 
/9 = a* = 78 = Sc = €7, 
<y = Sa = a€ = 6/8 = ^S, 

5 = ea = 07 = 7/3 =/8€, 

6 =:7a = aS=: S/8 =)87. 



368 A MEMOIR UPON CAUSTICS. [145 



XXIX. 

The preceding formulae, which were first given by me in the Philosophical Magdzine, 
December 1853, [124] include as particular cases a preceding theorem with respect 
to the caustic by refraction of parallel rays, and also two theorems of St Laurent, 
Oergonne, t. xviii., [1827, pp. 1 — 19] viz. if we suppose first that a=^Cy Le. that the 
radiant point is in the circumference of the refracting circle, then the system (a) shows 

that the same caustic would be obtained by writing c, - , 1 (or what is the same 

thing — 1) in the place of c, c, /x, and we have 

Theorem. The caustic by refraction for a circle when the radiant point is in the 
circumference is also the caustic by reflexion for the same radiant point, and for a 
reflecting circle concentric with the refracting circle, but having its radius equal to the 
quotient of the radius of the refracting circle by the index of refraction. 

Next, if we write a = CfjLj then the refracted rays all of them pass through a point 
which is a double point of the secondary caustic, the entire curve being in this case 
the orthogonal trajectory, not of the refracted rays, but of the false refracted rays; the 

formula (S) shows that the same caustic is obtained by writing -, c, 1 (or what is 

Cv 

the same thing — 1) in the place of a, c, /i f = - j , and we have 

Theorem. The caustic by refraction for a circle when the distance of the radiant 
point from the centre is to the radius of the circle in the ratio of the index of 
refraction to unity, is also the caustic by reflexion for the same circle considered as 
a reflecting circle, and for a radiant point the image of the former radiant point 



XXX. 

The curve is most easily traced by means of the preceding construction ; thus if 
we take the radiant point outside the refracting circle, and consider fi as varying fix)m 
a small to a large value (positive or negative values of /i give the same curve), we 
see that when fi is small the curve consists of two ovals, one of them within and 
the other without the refracting circle (see fig. 14). As /i increases the exterior oval 
continually increases, but undergoes modifications in its form; the interior oval in the 
first instance diminishes until we arrive at a curve, in which the interior oval is reduced 
to a conjugate point (see fig. 15); then as fi continues to increase the interior oval 
reappears (see fig. 16), and at last connects itself with the exterior oval, so as to 
form a curve with a double point (see fig. 17); and as /i increases still further the 



145] 



A MEMOIR UPON CAUSTICS. 



369 



curve again breaks up into an exterior and an interior oval (see fig. 18) ; and thence- 
forward as fjL goes on increasing consists always of two ovals; the shape of the exterior 
oval is best perceived fh)m the figures. An examination of the figures will also show 
how the same curves may originate from a different refracting circle and radiant point. 



Fig. 14. 



Fig. 17. 




Fig. 15. 





Fig. 18. 





0. II. 



47 



370 A MEMOIB UPON CAUSTICS. [l45 



XXXI. 

The theorem, " If a variable circle have its centre upon a circle S, and its radius 
proportional to the tangential distance of the centre from a circle (7, the envelope is 
a Cartesian," 
is at once deducible from the theorem — 

'' If a variable circle have its centre upon a circle S and its radius proportional 
to the distance of the centre from a point C\ the locus is a Cartesian,'* 

which last theorem was in eflfect given in discussing the theory of the secondary 
caustic. In fact, the hx^us of a point P such that its tangential distances from the 
circles C, C" are in a constant ratio, is a circle S, Conversely, if there be a circle C, 
and the locus of P be a circle S, then the circle C" may be found such that the 
tangential distances of P from the two circles are in a constant ratio, and the circle 
C may be taken to be a point, i.e. if there be a circle C and the locus of P be 
a circle S, then a point C may be found such that the tangential distance of F 
from the circle C is in a constant ratio to the distance from the point C\ 

Hence treating P as the centre of the variable circle, it is clear that the variable 
circle is determined in ihe two cases by equivalent constructions, and the envelope is 
therefore the same in both cases. 



XXXII. 

The equatiiJii of the secondary caustic developed and reduced is 
^* (x'' + yy - 2/x* (a*-* -\-{fi^+l) c^) (a^ + y^) + 8cV(W 4- a* - 2aV (/x^ + 1 ) + (/x^ - 1 )- c* = ( 
or, what is the same thing, 

which niav also be written 



which is of the form 

{x" 4- 3/' - ay -h 16.4 (./■ - m) = : 

and the values of the coefficients are 



m 



^,^ + 0+^)a\ 



145] 



A MEMOIR UPON CAUSTICS. 



371 



The equation just obtained should, I think, be taken as the standard form of the 
equation of the Cartesian, and the form of the equation shows that the Cartesian may 
be defined as the locus of a point, such that the fourth power of its tangential 
distance from a given circle is in a constant ratio to its distance from a given line. 



XXXIII. 

The Cartesian is a curve of the fourth order, symmetrical about a certain line 
which it intersects in four arbitrary points, and these points determine the curve. 
Taking the line in question (which may be called the axis) as the axis of x^ and a 
line at right angles to it as the axis of y, let a, 6, c, d be the values of x corre- 
sponding to the points of intersection with the axis, then the equation of the curve is 

# 

+ (.T - a) (x -b)(x- c) (x-d) = 0. 

It is easy to see that the form of the equation is not altered by writing x-\-0 for x, 
and a-\-0y b-\-0, c + O, d-\-0 for a, 6, c, d, we may therefore without loss of generality 
put a-\-b -\- c + d = 0, and the equation of the curve then becomes 



3/* + y2 (2^ + aft + ac + ad -h 6c -h M + cd) + (a: - a) (a? - 6) (x - c) (a; - d) = 0, 



where 



a-\-b-\-c-\-d = 0', 
the cur\'e is in this case said to be referred to the centre as origin. 

The last-mentioned equation may be written 

(^ + y^y -\-(ab-\-ac + ad-\-bc-\-bd-{-cd){af^ + j/^)- (abc -h aid + acd -f- bed) x + abed 



= 0, 



or 



[a^ -\- y"" -{-^ {db +ac + ad-\-bc -\-bd -\-cd)]' 
— (abc + ahd + acd -h bed) x 



-ii 



= 0, 



or 



( a^¥ + aV + a^'d^ + b^(^ + h'd? + c^d? 
+ 2a26c + 2a26d -h 2a^cd + 26«ac + Wad -h 2b^ed 
+ 2c«a6 + 2c2ad -h 'Id'bd + id'ah -h ^d'ac -h 2d'6c 
^ +2a6cd 
observing that 

a^be + a'hd H- a^'ed H- b^ae H- 6 W -h Ifed 
+ c^'ab + e^ad + e^'bd 4- d^ab + d^ac -h d?he 

= a6c (a + 6 + c) H- abd {a-\-b-\-d)-\- acd {a -\- e -{• d) -\-bed{b ^- c ■\- d) 



372 A MEMOIR UPON CAUSTICS. [l45 

the equation becomes 

j^i-" + y' + i (a6 + ac + od + 6c + M + cd)}« 

— (abc + abd + acd + bed) x 

- \ (a«6« + a^-\-d^d^ + 6«c« + 6«d* + c»d*- 6a6cd) = 0, 
which is of the fonn 

(j?* + y« - a)» + 16-4 (a? - 7^1) = 0, 
and, as already remarked, signifies that the fourth power of the tangential distance 



a point in the curve from a given circle, is proportional to the distance of the sai 
point from a given line. The circle in (juestion (which may be called the dirige^^^ 
circle) has for its equation 

^ + y' + i («^ + «c + ad + 6c + 6d + cd) = ; 

the line in question, which may be called the directrix, has for its equation 

d}V + a^c^ + a«d« + 6'c^ + h^d? + Cd* - 6a6cd ^ 
/p j_ — . f I « 

4 (oic + abd + ojcd + hcd) ' 

the multiplier of the distance from the directrix is 

abc + abd + acd + hcd. 

It may be remarked that a, 6, c, d being real, the dirigent circle is real; the equation 
may, in fact, be written 

^ + y' = 4 [(a + &)' + (a + c)« + (a + d)« + (6 + c)» + (6 + d)» + (c + d)>]. 



XXXIV. 

Considering the equation of the Cartesian under the form 

(a?> + y' - (jLf + 164 (a: - m) = 0, 

the centre of the dirigent circle a;* + y' — a = must be considered as a real point, 
but a may be positive or negative, i.e. the radius may be either a real or a pure 
imaginary distance: the coefficients A, m must be real, the directrix is therefore a real 
line. The equation shows that for all points of the curve a? — m is always negative 
or always positive, according as il is positive or negative, i.e. that the curve lies 
wholly on one side of the directrix, viz. on the same side with the centre of the 
dirigent circle if 4 is positive, but on the contrary side if A is negative. In the 
former case the curve may be said to be an * inside * curve, in the latter an ' outside ' 
curve. If 7/1 = 0, or the directrix passes through the centre of the dirigent circle, 
then the distinction between an inside curve and an outside curve no longer exists. 
It is clear that the curve touches the directrix in the points of intersection of this 
line and the dirigent circle, and that the points in question are the only points of 
intersection of the curve with the directrix or the dirigent circle; hence if the 
directrix and dirigent circle do not intersect, the curve does not meet either the 
directrix or the dirigent circle. 



145] A MEMOIR UPON CAUSTICS. 373 



XXXV. 

To discuss the equation 

I write first y = 0, which gives 

for the points of intersection with the axis of x. If this equation has equal roots, 
there will be a double point on the axis of x, and it is important to find the 
condition that this may be the case. The equation may be written in the form 

(3, 0, -a, 12A, 3a»-48ilm$a?, 1)* = 0, 

the condition for a part of equal roots is then at once seen to be 

- (a« - l2Amy + (a» - 18il77ia + 54^2)' = ; 

or reducing and throwing out the factor ^', this is 

27 A^ + 2m (8m« -da) A- a« (m« - a) = 0. 

This equation will give two equal values for A if 

m» (8m' - 9ay + 27a' (m» - a) = 0, 
an equation which reduces itself to 

(4m' - Say = 0. 

4m' 
Hence, if 4wi' — 3a be negative, i.e. if a>-^, the values of A will be imaginary, 

»j 

4m' . 4m' 

but if 4m' — 3a be positive, or a < - .- , the values of A will be real. If a = —^ , 

then there will be two equal values of A, which in fact corresponds to a cusp upon 
the axis of x. Whenever the curve is real there will be at least two real points on 

the axis of x; and when a<-^, but not otherwise, then for properly selected values 

of A there will be four real points on the axis of x. 

Differentiating the equation of the curve, we have 

{(af^ + f-a)x + 4iA) dx + {a^-tif'- a) ydy = ; 

and if in this equation we put dx = 0, we find y=0, or ar'-fy' — a = 0, i.e. that the 
points on the axis of x, and the points of intersection with the circle a^-\-y^ — a = 0, 
are the only points at which the curve is perpendicular to the axis of x. To find 
the points at which the curve is parallel to the axis of a;, we must write (ir = 0, this 
gives 

(a;' + y' - a) X + 4^ = 0, 



374 



A MEMOIR UPON CAUSTICS. 



[145 



and thence 



ic* + y* — a = — 



4^ 



X 



and 



^ + a^ (a? - m) = : 



this equation will have three real roots if ^<-^=-, and only a single real root if 



27 



4mv 



4m 



A > -^=- ; for J. = -^ , the equation in question will have a pair of equal roots. It 

is easy to see that there is always a single real root of the equation which gives 
rise to a real value of y, Le. to a real point upon the curve; but, when the equation 
has three real roots, two of the roots may or may not give rise to real points upon 
the curve. 



XXXVI. 



It is now easy to trace the curve. First, when m = 0, or the directrix 

through the centre of the dirigent circle, the curve is here an oval bent in so as 

to have double contact with the directrix, and lying on the one or the other side of 
the directrix according to the sign of A, See fig. a. 



Fig. a. 



Fig. h. 





..' 



Fig. c. 



Fig. d. 





145] A MEMOIR UPON CAUSTICS. 375 

Next, when the directrix does not pass through the centre of the dirigent circle, 
it will be convenient to suppose always that m is positive, and to consider A as 
passing first from to oo and then from to — oo , i. e. to consider first the 

different inside curves, and then the different outside curves. Suppose a > —^ , the 

o 

inside curve is at first an oval, as in fig. 6, where (attending to one side only of 

the axis) it will be noticed that there are three tangents parallel to the axis, viz. 

one for the convexity of the oval, and two for the concavity. For A = -^ the two 

tangents for the concavity come together, and give rise to a stationary tangent (Le. a 

tangent at an inflection) parallel to the axis, and for ^ > ^ the two tangents for 

the concavity disappear. The outside curve is an oval (of course on the opposite side 
of, and) bent in so as to have double contact with the directrix. 

Next, if a = —^ , the inside curve is at first an oval, as in fig. c, and there are, 

as before, three tangents parallel to the axis : for A = -^=- , the tangents for the con- 

cavity of the oval come to coincide with the axis, and are tangents at a cusp, and 

for A > -g^^ the cusp disappears, and there are not for the concavity of the oval any 

tangents parallel to the axis. The outside curve is an oval as before, but smaller and 
more compressed. 

Next, a < -^ > m*, then the inside curve is at first an oval, as in fig. d, and 
there are, as before, three tangents parallel to the axis; when A attains a certain 

477l' 

value which is less than y , the curve acquires a double point ; and as A further 

increases, the curve breaks up into two separate ovals, and there are then only two 
tangents parallel to the axis, viz. one for the exterior oval and one for the interior 
oval. As A continues to increase, the interior oval decreases ; and when A attains 

a certain value which is less than -nfr* ^^^ interior oval reduces itself to a conjugate 

point, and it afterwards disappears altogether. The outside curve is an oval as before, 
but smaller and more compressed. 

Next, if the directrix touch the dirigent circle, i.e. if a = m\ Then the inside 
curve is at first composed of an exterior oval which touches the dirigent circle, and 
*)f an interior oval which lies wholly within the dirigent circle. As A increases the 
interior oval decreases, reduces itself to a conjugate point, and then disappears. The 
outside curve is an oval which always touches the dirigent circle, at first very small 
(it may be considered as commencing from a conjugate point corresponding to ^ = 0), 
but increasing as A increases negatively. 



376 A MEMOIR UPON CAUSTICS. [l45 

Next, when the directrix does not meet the dirigent circle, i.e. if a < m*. The 
inside curve consists at first of two ovals, an exterior oval lying without the dirigent 
circle, and an interior oval lying within the dirigent circle. As A increases the 
interior oval decreases, reduces itself to a conjugate point and disappeara The outside 
curve is at first imaginary, but when A attains a sufficiently large negative value, it 
makes its appearance as a conjugate point, and afterwards becomes an oval which 
gradually increases. 

Next, when the dirigent circle reduces itself to a point, i.e. if a = 0. The inside 
curve makes its appearance as a conjugate point (corresponding to ^ = 0), and as A 
increases it becomes an oval and continually increases. The outside curve comports 
itself as in the last preceding case. 

Finally, when the dirigent circle becomes imaginary, or has for its radius a pure 
imaginary distance, i.e. if a is negative. The inside curve is at first imaginary, but 
when A attains a certain value it makes its appearance as a conjugate point, and 
as A increases becomes an oval and continually increases. The outside curve, as in 
the preceding two cases, comports itself in a similar manner. 

The discussion, in the present section, of the different forms of the curve is not 
a very full one, and a large number of figures would be necessary in order to show 
completely the transition from one form to another. The forms delineated in the four 
figures were selected as forms corresponding to imaginary values of the parameters bj 
means of which the equation of the curve is usually represented, e.g. the equations in 
Section xxvin. 



XXXVII. 

It has been shown that for rays proceeding from a point and refracted at a 
circle, the secondary caustic is the Cartesian; the caustic itself is therefore the evohite 
of the Cartesian ; this affords a means of finding the tangential equation of the 
caustic. In fact, the equation of the Cartesian is 

{x'-i-y'- ay + 16^ (a? - m) = ; 

and if we take for the equation of the normal 

Zf+F7; + ^ = 0, 

(where f, t) are current coordinates), then 

: a? (ar* -f y^ - a) H- 4.4 
: 4ily, 



145] A MEMOIR UPON CAUSTICS. 377 

equations which give 

Z*7» (a;» + y» - a) = ^AZ'XY^ 
whence eliminating, we have 

{Z» + X (mZ' - AX*)Y + 7» (mZ^ - AXy - Z»P (oZ + 4ilZ) = 0, 

where if, as before, c denotes the radius of the refracting circle, a the distance of the 
radiant point from the centre, and fi the index of refraction, we have 



A = 



d'a 



The above equation is the condition in order that the line Xx + Fy H- Z = may be 
a normal to the secondary caustic (a;" + y' — a)'+16^ (a? — m) = 0, or it is the tangential 
equation of the caustic, which is therefore a curve of the class 6 only. The equation 
may be written in the more convenient form 



xxxvin. 

To compare the last result with that previously obtained for the caustic by 
reflexion, I write ft = — 1, and putting also c = l and Z = a (for the equation of the 
reflected ray was assumed to be Xx + Fy + a = 0), we have 

a = a« + 2, A^\a, m = 2^(l + 2a«), 

a,nd the equation becomes, after a slight reduction, 

* 

4a< + 4o'Z (2o» + 1 - Z») + (Z» + F») (2o» + 1 - X")" - 4ia?Y* (a« + 2 + IX) = 0, 
^which may be writteu 

(2o' + Z (2o« + 1 - Z»))« + F» ( - 40' + 1 - Sa'X - 2 (2a'' + 1) X' + X*) = ; 
this divides out by the fector (X + iy, and the equation then becomes, 

(Z» - Z - 2a')> + F» ((Z - ly - 4a') = 0, 
■w-hich agrees with the result before obtained. 



378 A MEMOIR UPON CAUSTICS. [145 



XXXIX. 

Again, to compare the general equation with that previously obtained for parallel 
rays refracted at a circle, we must write /a = t> c=1, a = x, Z=^k (for the equation 
of the refracted ray was taken to be Xx -f Yy + ^ = 0) ; we have then 

a^l-vk^-Vk'a^ A^^k'a^ m = ^ (l +(1 + i»)a*) , 
and, after the substitution, a = x . The equation becomes in the first instance 
Jfc« + 2*»Z |~ (!+(!+ k") a«) t« - i**az4 + {X' + F«) — (l + (1 + i») a«) k' - J**(f.Y«J' 

and then putting a = x , or, what is the same thing, attending only to the tenuN 
which involve a', and throwing out the constant factor k^, we obtain 

(X^ + F«)(.Y« - 1 - k'Y - 4^'«F* = 0, 
or 

x^(X^-i-i(f^y+Y^{X-}-i+k)(X-i''k)(x + i-k){X--i-k)-~ 0, 

which agrees with the former result. 



XL. 

It was remarked that the ordinary construction for the secondary caustic* could 
not be applied to the case of parallel rays (the entire curve would in fact pass off 
to an infinite distance), and that the simplest course was to measure the distance 
GQ from a line through the centre of the refracting circle perpendicular to tin 
direction of the rays. To find the ecjuation of the resulting curve, take the centre of 
the circle as the origin and the direction of the incident rays for the axis of jc\ let 
the radius of the circle be taken equal to unity, and let fi denote, as before, the 
index of refraction. Then if a, /3 arc the coordinates of the point of incidence of a 
ray, we have a'^ + y^=l. and considering a, )8 as variable parameters connected bv this 
equation, the retjuired curve is the envelope of the circle, 

Write now a = cos^, y8 = sin^, then multiplying the equation by —2, and writing 
1 + cos 20 instead of 2 cos^ 9, the equation becomes 

1 + cos 20 - 2fi^ (x' -b if - 2x cos - 2i/ sin 5 + 1) = 0, 
which is of the form 

^cos2^ + 5sin2(9 + C'cose + i)sin^ + A' = 0, 



145] 



A MEMOIB UPON CAUSTICS. 



379 



and the values of the coefficients are 

4 = 1, 
B = 0, 

c = v^. 

D = Vy. 

E = -2fi*(ai' + y*) - 2yit»+ 1. 

Substituting these values in the equation 

12 (il» + £») - 3 (C» + i)*) + 4£>1' 

- {27il (C* - D") + BiBCD - (72 (4« + £«) + 9 (C* + i)»)) JE^ + 8£>}' = 0, 

the equation of the envelope is found to be 

16 {(1 - /*» + /*«) - 0*' + ,*«) {a? + r/») + At* (a^ + y')*;' 



( 4 - 6/*' - 6/** + 4|t« 
- (6/** + 3/** + Qfif) (a?+y>)- 27/*« (a? - yO 

-(6/*« + 6At')(«' + 2/')' 



^ > 



S-=o, 



which is readily seen to be only of the 8tb order. But to simplify the result, write 
iirst (a? + y*-l) + l, and 2*»-l -(«' + y*-l) in the place of a? + y* and a?-f respec- 
tively, the equation becomes 

4{(1 -/*')»- ft* (1 -/*») (a;* + y» - 1) + /** (a;* + y* - 1)"}* 
' 2(1 -/*•)» 
- 3/*" (1 - A*')* («^ + y* - 1) - 27/**a? 



— -< 



\ 



-3M*(i-/*»)(a!?+y-i)» 

+ 2/*« (a^ + y» - 1)» 



>• =0. 



/ 



Write for a moment 1— /a' = 5, ft*(.'c* + y'— l) = p, the equation becomes 
or developing, 

+ 54 (2g» - 39«p - Sgp* + 2/o») ;A*a;« - 72 Var* = 0, 
and reducing and dividing out by 27, this gives 

9«P»(P-5)» + 2(p + })(2p-})(p-23);aV-27;aV = 0, 



48—2 



380 A MEMOIR UPON CAUSTICS. [145 

whence replacing q, p by their values, the required equation is 

+ 2 (/x« (a;* 4- ya) - 2;a« + 1) (2/^2 (a;» + y*) - /*« - 1) (/*« (a;« + y«) - 2 + ;a«) a;^ 
which is the equation of an orthogonal trajectory of the refracted rays. 

In the case of reflexion, ;a = — I, and the equation becomes 

4(a;« + y*-l)»-27a^=0. 
Comparing this with the equation of the caustic, it is easy to see, 

Theorem. In the case of parallel rays and a reflecting circle, there is a secondaT"^ 
caustic which is a curve similar to and double the magnitude of the caustic, t\ 
position of the two curves differing by a right angle. 



XLI. 

The entire system of the orthogonal trajectories of the refracted rays might 
like manner be determined by finding the envelope of the circle (where, as befo: 
a, )8 are variable parameters connected by the equation a"-f )9* = 1), 

{The result, as far as I have worked it out, is as follows, viz. — 

(3 - 12 [m^ + 'Zmfi^x + ;a* (a:* + y')] + [1 - 2;a« + 2m* - 2/x« {a^ + y»)]*)» 
- ([1 - 2/A« + 2m» - 2/x» (a^ + f)] [9 + 18ah« + 36m/A«a: + 18;a* {a^ -h f)] 

- 54 [m» + imfi^x -^ti^ia^- /)] - [1 - 2;a« + 2iii» - 2;a« {a^ + y»)]»)* = 0, 
which, it is easy to see, is an equation of the order 8 only. Added Sept. 12. — A. C.j 



146. 



A MEMOIK ON CUKVES OF THE THIRD ORDER. 



[From the Philosophical Transactions of the Royal Society of London, vol. CXLVII. for the 
year 1857, pp. 415 — 446. Received October 30, — Read December 11, 1856.] 



A CURVE of the third order, or cubic curve, is the locus represented by an 
equation such as U={*^x, y, ^)» = 0; and it appears by my "Third Memoir on 
Quantics," [144], that it is proper to consider, in connexion with the curve of the third 
order 17=0, and its Hessian J? 17=0 (which is also a curve of the third order), two 
curves of the third class, viz. the curves represented by the equations PU=0 and QU=0. 
These equations, I say, represent curves of the third class; in fact, PU and QU are 
contravariants of J7, and therefore, when the variables x, y, z o{ U are considered as 
point coordinates, the variables f, 17, (f of PJ7 and QU must be considered as line 
coordinates, and the curves will be curves of the third class. I propose (in analogy 
with the form of the word Hessian) to call the two curves in question the Pippian 
and Quippian respectively. [The curve PU=0 is now usually called the Cayleyan.] 
A geometrical definition of the Pippian was readily found; the curve is in fact Steiner's 
curve i2o mentioned in the memoir "AUgemeine Eigenschafben der algebraischen Curven," 
Crelle, t. XLVii. [1854] pp. 1 — 6, in the particular case of a basis-curve of the third 
order; and I also found that the Pippian might be considered as occurring implicitly 
in my "M^moire sur les courbes du troisifeme ordre," Liouville, t. IX. [1844] pp. 
285 — 293 [26] and "Nouvelles remarques sur les courbes du troisi^me ordre," Liouville, 
t. X. [1845] pp. 102 — 109 [27]. As regards the Quippian, I have not succeeded in 
obtaining a satisfactory geometrical definition ; but the search aft^r it led to a variety 
of theorems, relating chiefly to the first-mentioned curve, and the results of the investi- 
gation are contained in the present memoir. Some of these results are due to Mr 
Salmon, with whom I was in correspondence on the subject. The character of the 
results makes it difficult to develope them in a systematic order; but the results 
are given in such connexion one with another as I have been able to present them 



382 A MEMOIR ON CURVES OF THE THIRD ORDER. [l46 

in. Considering the object of the memoir to be the establishment of a distinct 
geometrical theory of the Pippian, the leading results will be found summed up in 
the nine different definitions or modes of generation of the Pippian, given in the con- 
cluding number. In the course of the memoir I give some further developments 
relating to the theory in the memoirs in Liouville above referred to, showing its 
relation to the Pippian, and the analogy with theorems of Hesse in relation to the 
Hessian. 



Article No. 1. — Definitions, <tc. 

1. It may be convenient to premise as follows: — Considering, in connexion with 
a curve of the third order or cubic, a point, we have : 

(a) The first or conic polar of the point. 

(b) The second or line polar of the point. 

The meaning of these terms is well known, and they require no explanation. 

Next, considering, in connexion with the cubic, a line — 

(c) The first or conic polars of each point of the line meet in four points, 
which are the four poles of the line. 

{d) The second or line polars of each point of the line envelope a conic, which 
is the lineO'polar envelope of the line. 

And reciprocally considering, in connexion with a curve of the third class, a line^ 
we have: 

{e) The first or conic pole of th^ line. 

(/) The second or point-pole of the line. 

And considering, in connexion with the curve of the third class, a point — 

(g) The first or conic poles of each line through the point touch four linefi, 
which are the four polars of the point. 

(h) The second or point poles of each line through the point generate a conic 
which is the point-pole lociis of the point. 

But I shall not have occasion in the present memoir to speak of these reciprocal 
figures, except indeed the first or conic pole of the line. 

The term conjugate poles of a cubic is used to denote two points, such that the 
first or conic polar of either of them, with respect to the cubic, is a pair of lines 
passing through the other of them. Reciprocally, the term conjugate polars of a curve 
of the third class denotes two lines, such that the first or conic piole of either of 
them, with respect to the curve of the third class, is a pair of points lying in the 
other of them. 



146] A MEMOIR ON CURVES OF THE THIRD ORDER. 383 

The expression, a ayzygetic cubic, used in reference to two cubics, denotes a curve 
of the third order passing through the points of intersection of the two cubics; but 
in the present memoir the expression is in general used in reference to a single cubic, 
to denote a curve of the third order passing through the points of intersection of 
the cubic and its Hessian. As regards curves of the third class, I use in the memoir 
the full expression, a curve of the third class syzygetically connected with two given 
curves of the third class. 

It is a well-known theorem, that if at the points of intersection of a given line 
with a given cubic tangents are drnwn to the cubic, these tangents again meet the 
cubic in three points which lie in a line; such line is in the present memoir 
termed the satellite line of the given line, and the point of intersection of the two 
lines is termed the satellite point of the given line; the given line in reference to 
its satellite line or point is termed the primary line. 

In particular, if the primary line be a tangent of the cubic, the satellite line 
coincides with the primary line, and the satellite point is the point of simple inter- 
section of the primary line and the cubic. 



Article No. 2. — Oroup of Theorems relating to the Conjugal Poles of a Vubic, 

2. The theorems which I have first to mention relate to or originate out of the 
theory of the conjugate poles of a cubic, and may be conveniently connected together 
and explained by means of the accompanying figure. 

The point ^ is a point of the Hessian ; this being so, its first or c(mic polar, 
with respect to the cubic, will be a pair of lines passing through a point F of the 




Hessian ; and not only so, but the first or conic polar of the point F, with respect 
to the cubic will be a pair of lines passing through E. The pair of lines through 



384 A MEMOIR ON CURVES OF THE THIRD ORDER. [146 

F are represented in the figure by FBA, FDC, and the pair of lines through E are 
represented by EC A, EDC, and the lines of the one pair meet the lines of the other 
pair in the points A, B, C^ D. The point 0, which is the intersection of the lines 
AD, BC, is a point of the Hessian, and joining EOy FO, these lines are tangents to 
the Hessian at the points E, F, that is, the points E, F are corresponding points of 
the Hessian, in the sense that the tangents to the Hessian at these points meet in 
a point of the Hessian. The two points E, F are, according to a preceding definition, 
conjugate poles of the cubic. 

The line EF meets the Hessian in a third point 0, and the points G, are 
conjugate poles of the cubic. The first or conic polar of 0, with respect to the cubic, 
is the pair of lines AOD, BOC meeting in 0. The first or conic polar of 0, with 
respect to the cubic, is the pair of lines QEF and Gfeff! meeting in C The four 
poles of the line EO, with respect to the cubic, are the points of intersection of the 
first or conic polars of the two points E and 0, that is, the four poles in question 
are the points jP, F, e, e'. Similarly, the four poles of the line FO, with respect to 
the cubic, are the points E, E, /, f. 

The line EF, that is, any line joining two conjugate poles of the cubic, is a tangent 
to the Pippian, and the point of contact V is the harmonic with respect to the points 
E, F (which are points on the Hessian) of G, the third point of intersection with 
the Hessian. Conversely, any tangent of the Pippian meets the Hessian in three 
points, two of which are conjugate poles of the cubic, and the point of contact is the 
harmonic, with respect to these two points, of the third point of intersection with 
the Hessian. 

The line GO in the figure is of course also a tangent of the Pippian, and more- 
over the lines FBA, FDC (that is, the pair of lines which are the first or conic polar 
of E) and the lines EGA, EDB (that is, the pair of lines which are the first or 
conic polar of F) are also tangents to the Pippian. The point E represents any 
point of the Hessian, and the three tangents through E to the Pippian are the line EFG 
and the lines EGA, EDB; the line EFG is the line joining E with the conjugate 
pole F, and the lines EGA, EDB are the first or conic polar of this conjugate pole 
F with respect to the cubic. The figure shows that the line EG (the tangent to 
the Hessian at the point E) and the before-mentioned three lines (the tangents 
through E to the Pippian), are harmonically related, viz. the line EG the tangent of 
the Hessian, and the line EF one of the tangents to the Pippian, are harmonics 
with respect to the other two tangents to the Pippian. It is obvious that the 
tangents to the Pippian through the point F are in like manner the line GFE, and 
the pair of lines FBA, FBG, and that these lines are harmonically related to FO the 
tangent at F of the Hessian. And similarly, the tangents to the Pippian through 
the point are the line GO and the lines AOD, BOO, and the tangents to the 
Pippian through the point G are the line GO and the lines GFE and Gfefe', Thus 
all the lines of the figure are tangents to the Pippian except the lines EG, FO, 
which are tangents to the Hessian. It may be added, that the lineo-polar envelope 
of the line EF with respect to the cubic is the pair of lines OE, OF, 



146 J A MEMOIR ON CURVES OF THE THIRD ORDER. 385 

It will be presently seen that the analytical theory leads to the consideration of 
a line IJ (not represented in the figure): the line in question is the polar of E 
(or F) with respect to the conic which is the first or conic polar of F (or E) with 
respect to any syzygetic cubic. The line /J" is a tangent of the Pippian, and more- 
over the lines EF and // are conjugate polars of a curve of the , third class 
syzygetically connected with the Pippian and Quippian, and which is moreover such 
that its Hessian is the Pippian. 



Article Nos. 3 to 19. — Analytical investigations, comprising the proof of the 

theoremSy Article No. 2. 

3. The analytical theory possesses considerable interest. Take as the equation of 
the cubic, 

Z7 = ir' + y* + '^*+ ^locyz = ; 

then the equation of the Hessian is 

HU= f(af + f + z')-(\ + W) xyz^O; 

and the equation of the Pippian in line coordinates (that is, the equation which 
expresses that ^x+r)y-^ ^z = is a tangent of the curve) is 

Pir= - Z(|» -h 7;»+ ?») + (- 1 + 4P) ?i7?= 0. 
The equation of the Quippian in line coordinates is 

Qir= (1 - lOP) (f -h 7;» + ?») - 61^ (5 + 4P) fi;C= ; 
and the values of the two invariants of the cubic form are 

T=1-20P-8Z«, 
values which give identically, 

2^-64fll* = (l+8Z'')»; 

the last-mentioned function being in fact the discriminant. 

4. Suppose now that (X, Y, Z) are the coordinates of the point E, and 
(X\ Y, Z') the coordinates of the point F; then the equations which express that 
these points are conjugate poles of the cubic, are 

XT-{-l(YZ'^-rZ) =0, 

7F-hi(ZX'+^Z) = 0, 

ZZ' +1{XY + X'Y)=0; 

and by eliminating from these equations, first (X\ Y, Z\ and then (X, F, Z), we find 

ZH^* + F» + Z»)-(l+2P)XFir =0, 
p (X» + F» + Z'O - (1 + 2P) X'YZ' = 0, 
which shows that the points Ey F are each of them points of the Hessian. 



C. II. 



49 



386 A MEMOIR ON CURVES OF THE THIRD ORDER. [l46 

5. I may notice, in psussing, that the preceding equations give rise to a somewhat 
singular imaymmetrical quadratic transformation of a cubic form. In hct, the second 
and third equations give X' : T : Z'^YZ-PX^ : PX7-l^ : PZX-lYK And sub- 
stituting these values for X\ Y, Z' in the form 

Z«(Z'»+ F» + Z'»)-(l + 2l')X'rZ', 
the result must contain as a factor 

f(Z»+ F» + Z») - (1 + 2P) XTZ\ 

the other &ctor is easily found to be 

-P(P(Z«+F» + 2*) + 3ZZFZ). 

Several of the formulsB given in the sequel conduct in like manner to unsymmetrical 
transformations of a cubic form. 

6. I remark also, that the last-mentioned system of equations gives, symmetricaUy, 

X'* : y : Z" : FZ' : Z'X' : X'T 

= YZ-l*X* : ZX-l'Y^ : XT-PZ* : 1*YZ-IX* : 1*ZX-IY* : PXY-W; 

and it is, I think, worth showing how, by means of these relations, we pass from 
the equation between X', Y', Z! to that between X, Y, Z. In fact, representing, for 
shortness, the foregoing relations by 

X'* : T* : Z* : TZ' : Z'X' : X'T ^ A . B : G : F : Q : H, 
we may write 

X' = AF=GH, T = BQ = HF, Z' = CH = FQ, ABC = FGH; 
and thence 

X'*='AF.G'H\ T* = BO.H*F^, Z'* = CH.F>G', X'TZ^F'O'H*; 
hence 

l^iX'* + T* + Z*)-(l+2l*)X'TZ' = FGH{l^iAQH+BHF+CFQ)-(l + 2i')FOH}. 
But we have 

l^(AOH + BHF+ CFG) = - (21' + 1») (X* + F* + Z») XYZ+{1* + 2^)(Y*Z* + Z'X* + X*Y*), 
-(l + 2lf)FGH = (^ + 2l!>)(X*+Y* + Z*)XYZ+(l*+2P)(Y*Z* + Z*X* + X*Y') 

+ l'(l-l^)(l + 2l*)X'Y*Z'; 
and thence 

P (AGH+BHF+ CFG) - (1 + 21') FGH 

= _ p(l _ P) {P(Z«+ Y*+Z*)XYZ- (1 + 2l^)X'Y*Z'} ; 
and finally, 
l'(X"+ Y'* + Z'')-(l + 2l')X'Y'Z' = l'{-l + l*)(lYZ-X'){lZX- Y*)(IXY-Z')XYZ 

>i{l*(X'+Y*-\-Z*)-{l+2fi)XYZ\. 



146] A MEMOnt ON CURVES OF THE THIRD ORDER. 387 

We have also, identically, 

ABG-FQH = j(-l+l*)XYZ{l*(X*-\-Y* + Z>)-(l + 2l*)XYZ}, 
which agrees with the relation ABG — FGH=0. 

7. Before going further, it will be convenient to investigate certain relations 
which exist between the quantities (X, Y, Z), (X', Y, Z), connected as before by 
the equations 

XX' + liYZ' +TZ) =0, 

YT + 1 {ZX' + Z'X) = 0, 

ZZ' +i(XF' + Z'F) = 0, 

and the quantities 

^=YZ'-TZ, a=XX' = -j(YZ + TZ), 

v=zx'-z'x, fi=Yr=-jizx'+z'X), 

r = Zr - Z'F, y=ZZ' =-- (XT + X'Y). 

We have identically, 

2XX'(YZ'-TZ) + (XT + X'Y)(ZX'-ZX) + {ZX' + Z'X)(XT-X'Y) = 0; 

or expressing in terms of f, 17, ^, a, /9, 7 the quantities which enter into this 
equation, and forming the analogous equations, we have 

2K- fV- /9r=0, (A) 

-y^+2lfir,- ar=0, 

-/S|- o,, + 2tyi:=0. 
We have also 

X*rZ-X''YZ = i{-(XT + X'Y)(ZX'-Z'X) + {ZX' + Z'X)(XY'-X'Y)}, 

and thence in like manner, 

X»rZ' - X"YZ = ^ (7^ - /30, (B) 

Z*X'r - X'*YZ = i (j8f - ar,l 
Again, we have 

( YZ' - rzy ={YZ'+ YZf - 4 ytzz', 

(ZX' -Z'X)(Xr - X'Y)^-(ZX' + ZX)(Xr + X'Y) + 2XX' (YZ' + TZ); 

49—2 



388 A MEMOIR ON CURVES OF THE THIRD ORDER. [l46 

and thence 

p= i«'-4/87, (C) 

1;*= |-,/3»-4ya, 

and conversely 

J(l + 8P)«'= ?-4?.;r. (D) 

-l(l + 8P)7a=2V+ ?f. 
-J,(l + 8P)a/9 = 2ZC»+ f,. 

8. It is obvious that 

ix + fiy + J^ = 

is the equation of the line EF joining the two conjugate poles, and it may 
shown that 

aa? + /3y + 7-? = 

is the equation of the line /J", which is the polar of E with respect to a con ^ 
which is the first or conic polar of F with respect to any syzygetic cubic. In fitc^ 
the equation of a syzygetic cubic will be aj' + y*+^H-6Xa;y-? = 0, where X is arbitrar]!^ 
and the equation of the line in question is 

or developing, 

XTx+YYy + ZZ'z 
•\-\{YZ''^YZ)x-\-{ZT^Z'X)y + {XT + TY)z]^0\ 



146] A MEMOIR ON CURVES OF THE THIRD ORDER. 389 

and the function on the left-hand side is 

which proves the theorem. 

9. The equations (A) by the elimination of (f, % f), give 

-Z(a» + /S»+7») + (-H-4Z») 0^7=0, 

which shows that the line IJ is a tangent of the Kppian : the proof of the theorem 
is given in this place because the relation just obtained between a, /9, 7 is required 
for the proof of some of the other theorems. 

10. To find the coordinates of the point in which the line EF joining two 
conjugate poles again meets the Hessian. 

We may take for the coordinates of G, 

and, substituting in the equation of the Hessian, the terms containing u^ 1^ disappear, 
and the ratio u \ v \b determined by a simple equation. It thus appears that we 
may write 

V = 3P(Z»Z' -f Y^T + Z^Z')-{\ -f 2P) {YZT + ZXY -h XYZ') ; 
hence introducing, as before, the quantities f, 17, f, a, /9, 7, we find 

uZ + vX = 3P (717 - )80 + (1 + 2P) (-ST'FZ' - Z'«F^ ; 
but from the first of the equations (B), 

and therefore the preceding value of uX-^-vX' becomes 
which is equal to 

—21— (y't - ^^^- 

Hence throwing out the constant factor, we find, for the coordinates of the point (?, 
the values 

71; -^f, af-7f, ^f-ai7. 

11. To find the coordinates of the point 0. 

Consider as the point of intersection of the tangents to the Hessian at the 
points E, Fy then the coordinates of are proportional to the terms of 



3PZ» - l-f- 2PFZ , ZM^ - l-f- n^ZX , 3?2? - 1 + 2PZF 



3PZ'«-H-2Z»F^', 3PF«-1-h2Z»Z'Z', 3PZ'«- 1 -f 2PZ'F' 



390 A MEMOIR ON CURVES OF THE THIRD ORDER. [146 

Hence the a;-coordinate is proportional to 



which is equal to 

9i*(F»Z'«-F«2?)-l-3P(l + 2P)FF(XF-Z'r) + 3P(l + 2P)ZZ'(^Z'-Z'Z) 

-(1 + 2P)» ZZ'(FZ'-rZ); 

or introducing, as before, the quantities f, 17, f, a, /3, 7, to 

- 9Paf + 3P (1 + 2P) (/9?+ w) - (1 + Wy of, 
= (- 1 - 13/» - «•) af + 2Z« (1 + 2Z») (/9? + 71;). 

But by the first of the equations (A) /3f + 71; = 2Zaf , and the preceding value thus 
becomes ( — 1 — 7P + 8P) af. Hence throwing out the constant factor the coordinates of 
the point are found to be 

af fiVy 7?- 
12. The points (?, are conjugate poles of the cubic. 

Take a, 6, c for the coordinates of G, and a', h\ c* for the coordinates of 0, we have 

a, 6, c =r7-^?, a?-7f /3f-a^, 
a', 6', C = af , /9i7 , 7?. 

These values give aa' + i (6c' + 6'c) 

= «f(r7-/8?) + M/8'7(/8f-«'7) + 7?K-7f)} 

= ^7(«7 + W + '7"(~^«^) + r*(M + f?(-«/8-V); 

or substituting for f?/, 17', f ^ f f their values in terms of a, /9, 7, this is 

+ ( f -47a)( -/a/8) 
+ ( ^ -4ay9)( /a7) 

+ (-f/9«-ia7)(~«/8-V), 

which is identically equal to zero. Hence, completing the sjrstem, we find 

aa' + i (6c' + 6'c) = 0, 
W -h I {ca' + c'a) = 0, 
co' + i (a6' + a'6) = 0, 

equations which show that (as well as G) is a point of the Hessian, and that the 
points Qy are corresponding poles of the cubic. 



146] A MEMOm ON CUBVES OF THE THIRD OBDEB. 391 

13. The line EF joining a pair of conjugate poles of the cubic is a tangent of 
the Pippian^ 

In fact, the equations (A), by the elimination of a, /8, 7, give 

-f(P + ^*+?»)-l-(-l+4P)^7?=0, 
which proves the theorem. 

14. To find the equation of the pair of lines through F, and to show that these 
lines are tangents of the Pippian. 

The equation of the pair of lines considered as the first or conic polar of the 
conjugate pole E, is 

Z (a;» + nyz) + Y{y^ + 2lzx) + Z(z* + 2lxy) = 0. 

Let one of the lines be 

Xa? -f A^y + v^ = 0, 
then the other is 



X Y Z ^ 



and we find 



2lXfiv - Fi/» - Z/i« = 0, 
- Xv" + 2lYv\ - ^« = 0, 

- Z/A« - yx« + 2iZfiv = 0, 

any two of which determine the ratios X, /a, 1/. 

The elimination of Z, F, Z gives 

2Z/A1;, -i/», - /A« =0, 

- v" , 2,lv\ - X« 

- /A« , - XS 2iX/i 
which is equivalent to 

X/ii; { -Z(X» + /i« + !;») + (- 1 + 4P) X/ii/} = 0; 
or, omitting a factor, to 

- /(X» + /i« + !/») + (- 1 +4i»)X/ii; = 0, 

which shows that the line in question is a tangent of the Pippian. 

15. To find the equation of the pair of lines through 0, 

The equation of the pair of lines through ^ is in like manner 

Z'(a;* + 2iy2r)+ F(y« + 2i«a?) + Z'(^« + 2Ziry) = 0; 

^ Steiner's onrre Rf^ in the partioalar oase of a onbio baais-onrve, is aooording to definition tiie envelope 
of the line EF^ that is, the onrve R^ in the partionlar case in question is the Pippian. 



392 A MEMOIIt ON CURVES OF THE THIRD ORDER. [l46 

and combining this with the foregoing equation, 

X{a? + 2lyz) -f F(y' -f 2lzx) + Z(z^ + 2lxy) = 
of the pair of lines through F, viz. multiplying the two equations by 

X^X' -h PF 4- Z^Z\ - {XX'^ + YT^ + ZZ'% 
and adding, then if as before 

we find as the equation of a conic passing through the points A, B, C, D, the equation 

a (ic» + 2lyz) + h{y^ + 2lzx) + c{z'^ + 2lxy) = 0. 
But putting, as before, 

then a', 6', c' are the coordinates of the point 0, and the equations 

aa' + i(6c' + 6'c) = 0, 
66' + /(ca'+c'a) = 0, 

show that the conic in question is in fact the pair of lines through the point 0. 

16. To find the coordinates of the point F, which is the harmonic of with 
respect to the points E, F. 

The coordinates of the point in question are 

uX-vT, uY-vT, uZ-vZ', 

where u, v have the values given in No. 10, viz. 

u = -3i»(ZZ'«+ 77'« + ZZ'0 + (l + 2P)(FZ'X + ^Z'F + X'rZ), 
V = 3Z« (Z»X' + PF' + Z^Z') - (1 + 2P) (YZX' + ZXY' + XYZ") ; 
these values give 

uX - vZ' = - 81' {2Z»Z'« + (Zr + Z' F) YY' + (ZZ' + Z'Z) ZZ] 

+ (l + 2l^){(XY' + X'Y)(XZ'-\-X'Z) + Xr(YZ'+Y'Z]] 
and therefore 

iiZ-t;Z' = -3p|2a«-?/37 ■f(l + 2i«)|iy97-Ja«. 

= J,(l+8?)(-Za> + /87); 
and consequently, omitting the constant factor, the coordinates of F may be taken to be 

-fa« + /97, -//3» + 7a, -V + o^- 



146] A MEMOIR ON CURVES OP THE THIRD ORDER. 393 

17. The line through two consecutive positions of the point F is the line EF. 

The coordinates of the point T are 

-•la^-^l3y, ^l^ + ya, ^Irf + a/S; 
and it has been shown that the quantities a, /9, 7 satisfy the equation 

- Z (a» -f i8» 4- 7») + (- 1 + 4P) a/97 = 0. 

Hence, considering a, /9, 7 as variable parameters connected by this equation, the 
equation of the line through two consecutive positions of the point F is 

I ^si(^ + {-l + 4l^)fiy, -3//S«-l-(-l+4P)7a, - SZ^-h (- 1 -h 4P) a/3 > = ; 

^, -2la , 7 , /3 

y, 7 , -2Z/9 . a 

z, /3 , OL , -2/7 

and representing this equation by 

Lx + My -{• Nz = 0, 
we find 

Z= (4P/37 - a») (- 3/a» + (- l-f- 4/0/87) 
+ (a/9+ 2/7') (- n^ + (- 1 + 4P) 7a) 
+ (07 + 2/^) (- 3^y^ +(-l-h4P)a/9); 

or, multiplying out and collecting, 

L = 3Za* -h (- 1 - 8/») a?Py -h (- 5/ + 8P) (a/8» -h a7») -f- (- 16/» + 16/») /3V ; 
but the equation 

- Z (a» -f- y3» -f- y) + ( - 1 + 4/^ 0/87 = 
gives 

3^0* = - 3i (ay8« + OT*) + (- 3 + 12i») tf»/97, 
and we have 

Z = (- 4 + 4P) OL^Py + (- 8i + SV) (o/S^-h ay) + (- 16P + 16Z0/3V 

= (- 4 + 4Z») (a»/37 + 2Z (a5» -h a^) + 4Z»/8V) 

= (_ 4 + 4i.) (a^ + 2/^) (a/9 -h 2ty») ; 

or, in virtue of the equations (D), 

Z = (-4 + 4P)Z«5f .Pfi7 = (-4 + 4P)i*fi7?=(-4-h4/»)/*fi;?.f. 

Hence, omitting the common factor, we find L : M : N=^ : r) : ^, and the equation 
Lx + My + Nz = becomes 

f a? + w + C^ = 0, 



C. II. 



50 



394 A MEMOIB ON CUBYES OF THE THIRD ORDER. [l46 

which is the equation of the line EF, that is, the line through two consecutive positioDs 
of r is the line EF\ or what is the same thing, the line EF touches the Pippian 
in the point F which is the harmonic of Q with respect to the points E, F. 

18. The lineo-polar envelope of the line EF, with respect to the cubic, is the 
pair of lines OE, OF. 

The equation of the pair of lines OE, OF, considered as the tangents to the 
Hessian at the points E, F, is 

r=0. 

X {(3PZ'« - 1 + 2PFZ0 X -f- {ZP r* - 1 + WZ'X') y + (3P^'* - 1 + 2PX'Y') z] ) 

Here on the left-hand side the coefficient of a;* is 

9/*Z«Z'« - 3? (l-f- 2V) (X'Y'Z' + X'^YZ) + (1 + 2P)« YTZ^, 
which is equal to 



that is 



91^ - 3P (1 + 2Z») (Z«/97 + ^ a«) + (1 -h 2Z»)» /Sy, 



j(-/ + ?){3ia»+2(l-h2P)/37}; 



and the coefficient of yz is 

9^(F«Z'»+ F«2?) -3Z«(1 + 2V) {YY' {XY' + X'Y)^ ZZ'iXZ' -^^X'Z)) 

-h(l+2P)«ZZ'(F^+F'Z), 
which is equal to 



that is 



J(-/-l-/*){(l-4/»)a«-6P/97) 



Hence completing the system and throwing out the constant factor, the equation of 
the pair of lines is 

(3k«-|-2(H-2Z»)/87, 3/i8« + 2 (1 + 2P) 7a, 3^ + 2 (1 -h 2Z') a/9. 

(1 » 4P) a« - 6/^/87, (1 - 4P) y3« - QPyOL, (1 - 4P) 7* - &Pol^\x, y, ^)« = 0. 

But the equation of the line EF is |a? -h ^y + f^ = 0, and the equation of its liueo-polar 
envelope is 



f, X, Iz. ly 
f), Iz, y, Ix 
?, ly, X , z 



= 0; 



146] A MEMOIR ON CUBVES OF THE THIBD OBDEB. 395 

or expanding, 

or arranging in powers of x, y, z, 

(-/«p-2/i7r, -tv-m. -^"^-2i?i7, ip+^;?, hv'+m> iP+mii^,y>^y^o: 

and if in this equation we replace f , &c. by their values in terms of a, fi, y, aa 
given by the equations (D), we obtain the equation given as that of the pair of lines 
OE, OF. 

19. It remains to prove the theorem with respect to the connexion of the lines 
EF, TJ. 

The equations (A) show that the two lines 

^x+Tfy +5e^ = 0, 
ax-\-fiy+^=0, 

(where f, i;, f and a, /3, y have the values before attributed to them) are conjugate 
polars with respect to the curve of the third class, 

in which equation |; 17, ^ denote current line coordinates. The curve in question is of 
the form APV +BQV = 0. We have, in fact, identically, 

It is clear that the curve in question must have the curve PCr = for its Hessian; 
and in fact, in the formula of my Third Memoir, [144] 

H{6aPU + fiQU)=^(-2T, 48S*, ISTS , T*+16S»$a, fiyPU 

+ ( 8>8f, r,-8S*, ^T8 Ha, fiYQU, 

the coefficient of QCT is 

and therefore, putting a = JT, ^ = — 4/S, we find 

JT(3r.Pcr-4S.<2J7) = -H3^-645*)»Pcr. 



Article No. 20. — Theorem relating to the curve of the third doss, mentioned in the 

preceding Article, 

20. The consideration of the curve ST.PU— 4aS.QU=0, gives rise to another 
geometrical theorem. Suppose that the line (^, 17, (f), that is, the line whose equation 
is ^ + rfy + ^z = 0, is with respect to this curve of the third class one of the four 
polars of a point {X, T, Z) of the Hessian, and that it is required to find the envelope 
of the line ^x + r)y+^z = 0, 

50—2 




396 A UEHOIB ON CURVES OP THE THIRD ORDER. [U6 

We have 

X: y,^-ip-,j;: V-ff^if-ft, 
and X, y, Z are to be eliminated from these equations, and the equation 

PiX*+V' + Z*)-{l+2l')XrZ=0 
of the Hessian. We have 

x'+Y'+z'= Hp+v'+^y 

+ 91 fVf 
-(l+2P)(i,'f + rP + ?-;*). 

xYz= i(p+y+c.)f,{: 
4 (-1+1')^^ 

-^(VS' + rP + P'j'). 

and thence 

nu= P (^+^> + ^y 

-(i + 5l^)(P + v' + n^? 

+ (I + 101' -21*)^^; 

and equating the right-hand side to zero, we have the equation in line coordinates of 
the curve in question, which is therefore a curve of the sixth class in quadratic 
syzygy with the Pippian and Quippian. 



Articl« No. 21. — Geometrical definition of the Quippian, 

21. I have not succeeded in obtaining any good geometrical definition of the 
Quippian, and the following is only given for want of something better. 



T. PU \PeH(aU + 6^HU)\ - P{6HU) [T{aU+&^HU) .P{aU+ Q^HU)] =0, 

which is derived in what may be taken to be a known manner from the cubic, is ii 
general a curve of the sixth class. But if the syzygetic cubic aU + 60HV = be 
properly selected, viz. if this curve be such that its Hessian breaks up into three 
lines, then both the Pippian of the cubic aU+6^HU = 0, and the Pippian of iU 
Hessian will break up into the same three points, which will be a portion of the 
curve of the sixth class, and discarding these three points the curve will sink down 
to one of the third class, and will in fact be the Quippian of the cubic. 

To show this we may take 

an + 60HU = iX!'+y' + A=O 



146] A MEMOIIt ON CURVES OF THE THIRD ORDER. 397 

as the equation of the syzygetic cubic satisfying the prescribed condition, for this value 
in fact gives 

H (aU-^efiHU) = - xyz, = 0, 

a system of three lines. We find, moreover, 

P(aU + 6fiHU) = P(x' + f+z'l:=''^^ 
and 

P {6H{aU-h 6l3HU)]^P{''6ayz), = -4,^^, 

the latter equation being obtained by first neglecting all but the highest power of I in 
the expression of PU, and then writing l = —l: we have also T(aU-\-6l3HU)=l. 
Substituting the above values, the curve of the sixth class is 

^^^^T.PU+P(6HU)]=0', 
or throwing out the factor fi/Jf, we have the curve of the third class, 

-^T.PU+P(6HU) = 0. 
Now the general expression in my Third Memoir, viz. 

P {aU+ 60HU) = (a» + USafi' + *T^) PU -¥ (c^fi - 48/3*) QU, 
putting a = 0, /8 = 1, gives 

P (6HU) = iT.PU-i8.QU, 
or what is the same thing, 

-4T.PU + P{6HU) = -4>8,QU; 

and the curve of the third class is therefore the Quippian QU = 0, It may be remarked, 
that for a cubic Cr = the Hessian of which breaks up into three lines, the above 
investigation shows that we have Pf/^= — fi?^, P(6-ffJ7) = — 4fi7f. and T=l, and conse- 
quently that --^T.PU -{- P{6HU) ought to vanish identically; this in fact happens in 
virtue of the factor 8 on the right-hand side, the invariant iS of a cubic of the form 
in question being equal to zero ; the appearance of the factor 8 on the right-hand 
side is thus accounted for d priori. 



Article No. 22. — Theorem relating to a line which meets three given conies in six points in 

involution. 

22. The envelope of a line which meets three given conies, the first or conic 
polars of any three points with respect to the cubic, in six points in involution, is 
the Pippian. 

It is readily seen that if the theorem is true with respect to the three conies, 

dx ' dy * dz * 



398 A M£MOIB ON CURVES OF THE THIBD ORDER. [l46 

it is true with respect to any three conies whatever of the form 

^ dU , dU dU ^ 

that is, with respect to any three conies, each of them the first or conic polar of 
some point (X, /a, v) with respect to the cubic. Considering then these three conies, 
take ^x+riy-\' ^z = as the equation of the line, and let (Z, F, ^ be the coordinates 
of a point of intersection with the first conic, we have 

fZ-fi7F+?Z=0, 
X^+21YZ =0; 

and combining with these a linear equation 

aX-{-l37+yZ=0, 

in which (a, /3, 7) are arbitrary quantities, we have 

and hence 

an equation in (a, /3, 7) which is in fact the equation in line coordinates of the two 
points of intersection with the first conic. Developing and forming the analogous 
equations, we find 

( V' > r , -2Za ^rf , Iv^ , -fiy-if'ia, A 7)" = 0, 
which are respectively the equations in line coordinates of the three pairs of intersectionB. 

Now combining these equations with the equation 7 = 0, we have the equations 
of the pairs of lines joining the points of intersection with the point (« = 0, y = OX and 
if the six points are in involution, the six lines must also be in involution, or the 
condition for the involution of the six points is 

-2^r, ^ , m =0. 

that is, 

4P|:«^; (- f7 - ^D + Vr + ^f?* + 2i»|Vr + Sf'lVC + C* (- f^ - ^D = ; 

or, reducing and throwing out the fsictor ^, we find 

-K?+v + r)+(-n-*i')fv?=o, 

which shows that the line in question is a tangent of the Pippian. 



146] A M£MOIB ON CURVES OF THE THIRD ORDER. 399 

It is to be remarked that any three conies whatever may be considered as the 
first or conic polars of three properly selected points with respect to a properly selected 
cubic curve. The theorem applies therefore to any three conies whatever, but in this 
case the cubic curve is not given, and the Kppian therefore stands merely for a curve 
of the third class, and the theorem is as follows, viz. the envelope of a line which 
meets any three conies in six points in involution, is a curve of the third class. 



Article No. 23. — Completion of the theory in Liouville, and comparison ivith analogous 

theorems of Hesse. 

In order to convert the foregoing theorem into its reciprocal, we must replace the 
cubic U=0 by a curve of the third class, that is we must consider the coordinates 
which enter into the equation as line coordinates ; and it of course follows that the 
coordinates which enter into the equation PU = must be considered as point 
coordinates, that is we must consider the Pippian as a curve of the third order: we 
have thus the theorem; The locus of a point such that the tangents drawn from it 
to three given conies (the first or conic poles of any three lines with respect to a 
curve of the third class) form a pencil in involution, is the Pippian considered as a 
curve of the third order. This in fact completes the fundamental theorem in my 
memoirs in Liouville above referred to, and establishes the analogy with Hesse s results 
in relation to the Hessian ; to show this I set out the two series of theorems as 
follows : 

Hesse, in his memoirs On Curves of the Third Order and Curves of the Third 
Class, CreUe, tt. xxviii. xxxvi. and xxxviii. [1844, 1848, 1849], has shown as follows : 

(a) The locus of a point such that its polars with respect to the three conies 
jr = 0, F=0, Z=0 (or more generally its polars with respect to all the conies of the 
series \X -^ /mY •{• pZ =^0) meet in a point, is a curve of the third order F = 0. 

(fi) Conversely, given a curve of the third order F=0, there exists a series of 
conies such that the polars with respect to all the conies of any point whatever of 
the curve F=0, meet in a point. 

(7) The equation of any one of the conies in question is 

dx dy dz * 

that is, the conic is the first or conic polar of a point (X, /a, v) with respect to a 
certain curve of the third order 17'= 0; and this curve is determined by the condition 
that its Hessian is the given curve F=0, that is, we have V=HU. 

(S) The equation V^HU is solved by assuming U = aV+bHV, for we have then 
H (aV'\-bHV)=^AV + BHV, where A, B are given cubic functions of a, 6, and thence 
V=sHU=AV+BHV, or -4 = 1,5 = 0; the latter equation gives what is alone important, 
the ratio a : b; and it thus appears that there are three distinct series of conica, 



400 A MEMOIR ON CURVES OF THE THIRD ORDER. [l46 

each of them having the above-mentioned relation to the given curve of the third 
order F=0. 

In the memoirs in Lioumlle above referred to, I have in effect shown that — 

(a') The locus of a point such that the tangents from it to three conies, repre- 
sented in line coordinates by the equations X = 0, F=0, Z = (or more generally with 
respect to any three conies of the series \X + fiT-\'vZ=0) form a pencil in involution, 
is a curve of the third order V=0. 

(^) Conversely, given a curve of the third order F = 0, there exists a series of 
conies such that the tangents from any point whatever of the curve to any three of 
the conies, form a pencil in involution. 

Now, considering the coordinates which enter into the equation of the Pippian as 
point coordinates, and consequently the Pippian as a curve of the third order, I am 
able to add as follows : 

(7') The equation in line coordinates of any one of the conies in question is 

^ dU dU dU ^ 

that is, the conic is the first or conic polar of a line (X, fi, v) with respect to a 
certain curve of the third class U = ] and this curve is determined by the condition 
that its Pippian is the given curve of the third order F = 0, that is, we have 
V=PU. 

(Sf) The equation F = Pf7' is solved by assuming U=aPV+bQVy for we have 
then P(aPV-\'bQV)==AV'^BHVy where A and B are given cubic functions of a, i; 
and thence V=^PU = AV+ BHV, or ^=1, 5 = 0; the latter equation gives what is 
alone important, the ratio a : b; and it thus appears that there are three distinct 
curves of the third class [7=0, and therefore (what indeed is shown in the Memoirs 
in Liouville) three distinct series of conies having the above-mentioned relation to the 
given curve of the third order F= 0. 

It is hardly necessary to remark that the preceding theorems, although precisely 
analogous to those of Hesse, are entirely distinct theorems, that is the two series are 
not connected together by any relation of reciprocity. 



Article Nos. 24 to 28. — VarioiLS investigations and theorems. 

24. Reverting to the theorem (No. 18), that the lineo-polar envelope of the line 
EF is the pair of lines OE, OF; the line EF is any tangent of the Pippian, hence 
the theorem includes the following one: 



146] A MEMOIR ON CURVES OF THE THIRD ORDER. 401 

The lineo-polar envelope with respect to the cubic, of any tangent of the Pippian, 
is a pair of lines. 

And conversely, 

The Pippian is the envelope of a line such that the lineo-polar envelope of the 
line with respect to the cubic is a pair of lines. 

It is I think worth while to give an independent proof. It has been shown that 
the equation of the lineo-polar envelope with respect to the cubic, of the line 
f J- -h lyy + Jf^ = (where f , i;, Jf are arbitrary quantities), is 

(_/.^_2f,f, -ihf-2i^. -p^-2i^, ir+^*^r. W+m if'+PM^. y. ^)'=o; 

and representing this equation by 

i (a, 6, c, /, g, h\x, y, zf = 0, 
we find 

he -/« = f (- r + 8ZV+ S^'?" + 12Pf«7?)> 

a6 - A« = ? (8Pf + 8ZY - ?* + 12f^^70> 

gh - a/= f {21' (^ + ^ + f 3) + 4; (1 + 2P) fi7?) + (1 + 8P) 7;«?«, 

A/-6i7 = ^(2P(r + ^+?^)+4Kl + 2P)f^?)-h(l + 8P)rr, 
/flr-cA = r(2/»(r + y + f') + «(l-h2?)^?)-h(l+8Z')r^^ 

and after all reductions, 

ahc-af'-hg'-ch?^2fgh 

or the condition in order that the conic may break up into a pair of lines is PU=0. 

25. The following formulae are given in connexion with the foregoing investigation, 
but I have not particularly considered their geometrical signification. The lineo-polar 
envelope of an arbitrary line ^x-{-r)y-\'^z=0, with respect to the cubic 

has been represented by 

(a, b, c, /, g, A$a?, y, ^)' = ; 

and if in like manner we represent the lineo-polar envelope of the same line, with 
respect to a syzygetic cubic 

a^ + y^ + z^-{-6Vxyz=:i), 

by 

(a\ h\ c\ /, g\ hj^x, y, zf = 0, 
c. II. 51 



402 



A MEMOIR ON CUEVBS OF THE THIRD ORDER. 



[146 



then we have 

a' (6c -/*) + b' (ca - g*) + c' {ab - h*) + 2f(gh- of) + 2g' {hf- hg) + 2h' (fg - ch) 

{l" + 2l*)(^ + r^ + ^y 
+ (21' + 4i - S21H' + 81*) (p + ff + n f7? 
+ (•24M'» + mH'^ - 72W' + 24P + 3) }^^, 

which may be verified by writing V = 1, in which case the right-hand side becomes 

1 + 2P 
it should do, 3{PUy. If I' = «/«" » *^** ^' ^^ ^^^ syzygetic cubic be the Hessiib 

then the formula becomes 



a'{bc-f) + &c.= 



361' 



(l + 4i» + 76Z«)(f + V + f»)» ^ 

+ 12f' ( - 1 + 26^' + 56P) (f + .;' + f») f»7? 
+ 121 (2 + 57^ + 168i' + 16^") fV(r* J 



>■ 



which is equal to 



afeK-^*^-^^'- 



26. The equation 

{bc' + h'c-2ff,...gh'+g'h-af-a'f,...1t 17, O' = 

is the equation in line coordinates of a conic, the envelope of the line which cut<i-^ 
harmonically the conies 

(a, b, c, /, g, h $a:, y, zf = 0, 
(a', 6', c', /. gf, h'\x, t/, zy = 0; 

and if a, b, &c., a', &c. have the values before given to them, then the coefficients 
of the equation are 

be' + b'c-2ff = f { - f + 4«' (Z + I') iff + ^) + iUW - 21' - 21'*) fv?, 
ca' + e'a-2gg' =i» {-ij» + 4«' (Z + (?* + ?) + (16«'-2P- 2Z'«)fi,?, 
ab' + a'b- 2hh' = f ( - f»+ 4M' (Z + (? + '?*) +(16«'- 2P- 2^ ^^ 

gh' + g'h -af - a'f= f {(Z" + /'•) (f> + ,» + f») + {21 + 21' + 8W'>) fij?) + (1 + 4tf' (Z + O) ly^. 
A/' + h'f- bg' -b'g=fi {(P + 1'*) (p + ,^ + (:») + (2Z + 2^ + 8W) fijf} + (1 + 4tt' (f + f)) f»f». 
/5^ +/i7 - cA' - c'A = r {(P + Z'') (P + 1;* + f) + (2Z + 2Z' + 8W'«) f^f } + (1 + 4«' (f + Z')) f»,» : 

and we thence obtain 

(bc' + b'c~2/f,..,gh'+g'h-af-a'/,..Jil r,, ?)• = 

+ ( f+ 1'*+ 16U') (? + ff+^)^^ 
+ (6l + 6l' -l-24W»)^f» 

+ (4 +16(W' + «'*))(,»?'+ f«f»+fS,'). =0 



146] A MEMOIR ON CURVES OF THE THIRD ORDER. 403 

{is the condition which expresses that a line fa? + lyy + f^ = cuts harmonically its 
lineo-polar envelopes with respect to the cubic and with respect to a syzygetic cubic. 

27. To find the locus of a point such that its second or line polar with respect 
to the cubic may be a tangent of the Pippian. Let the coordinates of the point be 
(a?, y, z) ; then if fa? + lyy + fer = be the equation of the polar, we have 

and the line in question being a tangent to the Pippian, 

But the preceding values give 

P + ^' + r' = («* + y' + '^)' + 6/(ar» + y» + ^)a?y2'-f- SGPaj^y^-^' -f ( - 2 + 8P) (y*^ + ^ar» -f ay ) 

fiyf = 4fP {aj" -\- y^ ■}- z^) xyz + (1 + Sl^) /c'y^z^ -^ 21 (y^z' + r^ar'-f ay); 

and we have therefore 

/ (a?" + y» + -?»)' + (10i« - 1 6^) (a;* + y' + -^') a:y2: -f- (1+ 40? - 32P) a^fz'' = ; 

or introducing IT, HJJ in place of a:* + y* + 2:^, xyz, the equation becomes 

which is the equation of the locus in question. 

28. The locus of a point such that its second or line polar with respect to the 
cubic is a tangent of the Quippian, is found in like manner by substituting the last- 
mentioned values of f , 17, f in the equation 

Q[7 = (l-10ZO(P + i;*+H-6Z«(5 + 4/*)fi75'. 

We find as the equation of the locus, 

(1 - 10i») (a;» + y» + -e»)> + 6i (1 - 30f» - 1 6/«) (a:* -h y* + ^) ay^ + 6i* (1 - 104Z' - 32/«) ji^y'z' 

-2(1-1- 8P)» (yV + z^a^ + a^f) = 0, 

where the function on the left-hand side is the octicovariant ^„U of my Third 
Memoir, the covariant having been in fact defined so as to satisfy the condition in 
question. And I have given in the memoir the following expression for 0„jr, viz. 

e„ir=(l-16f-6/«)f^* 

+ (6i )U .HU 

+ (6i' ){HUf 

2(1+ 8P)» {f2^ + «»a;» + aft/'). 

51—2 



404 A MEMOIR ON CURVES OF THE THIRD ORDER. [146 



Article Nos. 29 to 31. — Formulod for the intersection of a ciMc curve and a line. 

29. If the line ^x-^rjy -\- ^z = meet the cubic 

ic» + y»^ -f- 6lxyz = 
in the points 

then we have 

It will be convenient to represent the equation of the cubic by the abbreviate^ 
notation (1, 1, 1, l^x, y, 2^)^ = 0; we have the two equations 

(1, 1, 1, Z$^, y, ^)»=0, 

far-fiyy + f^ =0; 

and if to these we join a linear equation with arbitrary coefficients, 

aa -}- /3y -\- yz = 0, 
then the second and third equations give 

and substituting these values in the first equation, we obtain the resultant of the 
system. But this resultant will also be obtained by substituting, in the third equation, 
a system of simultaneous roots of the first and second equations, and equating to 
zero the product of the functions so obtained ^ We must have therefore 

(1, 1, 1, ?$^?-7i7, 7f-< a^ - /3f )' = (a^ + ^yi + 7^i) («^2 + ^ya + T-^j) (o^j + ^^3 + 7^s) ; 
and equating the coefficients of a', I3\ 7*, we obtain the above-mentioned relations. 

30. If a tangent to the cubic 

a^-\- y^-\-z^ -\- 6lxyz = 
at a point {x^, y^, z^) of the cubic meet the cubic in the point (^,, y,, ^), then 

«s : ys : -2^8 = a^ (yi» - z^*) : y^ (V - x^^) : z, (x^^ - yi>). 
For if the equation of the tangent is f^ + lyy + f^ = 0, then 

and 

^ : V ' S'=^i* + 2Zyi^, : yi^-\-2lziXi : z^^ + 2lxiyi. 

^ This is in fact the general process of elimination given in Sohlafli's Memoir, '* Ueber die Besultante 
einer Systemes mehrerer algebraischer Gleichongen," Vienna Trans. 1S52. [But the prooess was employed much 
earlier, by Poisson.] 



146] A MEMOm ON CURVES OF THE THIRD ORDER. 405 

These values give 

= (yi' - ^i') X - (1 + 8P) x,\ 

since (a^, yi, ^i) is a point of the cubic; and forming in like manner the values of 
f* — f and f — rj\ we obtain the theorem. 

31. The preceding values of (a?„ y„ z^) ought to satisfy 

(a^» + 2/y,2ri) a^ + (yi» -f 2h,x,) y, + (-^i* + 2lx^i) z^ = 0, 
^' + yj' + ^i + 6te,ys^, = ; 

in fact the first equation is satisfied identically, and for the second equation we 
obtain 

x^^ + y.' + ^3' = ^a' (y 1' - ZiJ + y,' (^,' - x,J + ^,» (a:,« - y,J 

= - ^1' (y,' - zi') - yi' (V - ^1') - V C^i' - yiO 
= (^i' + yi' + ^,') (yi' - z,^) (z,' - ^j») (a:i» - y,% 

and consequently 

x/ + y»' + z,' + 6/a?,y,2r, = (a^» + y,» + ^j» + 6lx^iZ,) (y,^ - z,^) ( ^ - x,^) {x,^ - y,') = 0. 

which verifies the theorem. It is proper to add (the remark was made to me by 
Professor Sylvester) that the foregoing values 

^3 : ys : ^j = a?, (yi» - z,*) : y, (z,^ - a?,') : z^ (x,^ - y,') 

satisfy identically the relation 

a?3^ + ys^ -H -^3' _ ^jM^yijff 2^ 
x^ysZa ^\y\Z\ 



Article Nos. 32 to 34. — Formulce foi* the Satellite line and point. 

32. The line fa? + lyy + 5^ = meets the cubic 

a^^y9^2^^ Qlxyz = 

in three points, and the tangents to the cubic at these points meet the cubic in 
three points lying in a line, which has been called the Satellite line of the given line. 

To find the equation of the satellite line; suppose that (xi, y^, z^), (a?j, y,, z^), 
(a^s, y3» Z3) are the coordinates of the point in which the given line meets the cubic; 
then we have, as before, 

(1, 1, 1, «5^?-7i;, 7f-af, ai? - /3f)« = (oa^ + /9y, + 7^i)(flw?a 4^ /3y2 + 7^a)(<MJ8 + y9y, + 7-^3). 



406 



A MEMOIR ON CURVES OF THE THIRD ORDER. 



[146 



The equation of the three tangents is 

n = [(x,^ + 2ly,z,) X + (yi> + 2lz,x,) y + {z^^ + 2/a?iy0 z'^ =0, 
X [(a:,« + 2ly^;) X + (y,« + 2lz^;) y + (V + 2ir^,) 2r] 

and if we put 

F=(f» + if + ?»)'-24?»(p + i7»+f«)fi;?+(-24Z-48Z*)fV?' + (-4 + 32i0(i;'f*+?'f + fV^ 
(^ is the reciprocant FU of my Third Memoir), then we have identically 

and the equation of the satellite line is fa? + 17'^ + 5^2^ = 0. In fact the geometrical 
theory shows that we must have 

and it is then clear that iV is a mere number. To determine its value in the most 
simple manner, write Z = 0, y = 0, a7=f, z = — ^, we have then F, 17—^11 = 0, where 

The value of IT is II = F . U, and we thus obtain iV=l. For, substituting the above 
values, 

-rf(aaVai.' + &c.) 
+ ?f»(a;,z,'z,' +&C.) 
- 1*^1 W, 



and we have 



and thence 



and consequently 



a^a;,«, + &c. = SJi'f, 
(Ci^,?, + &c. = - Sf^, 

x,Wz** + &c. = 9?*^ + 6?? » (y - r) = 3?«p + 6(?*i,», 
a;.VV + &c. = 9(7^* - 6?»f (f - '?') = 3^?* + ^^W' 

n= (7(17*- r)" 

- r(r-'?')' 

= (?*-?)(?+ 1?* + r - 2?''?' - Srp - 2pi;'). 



146] A MEMOIR ON CURVES OF THE THIRD ORDER. 407 

Now considering the equation 

in order to find f, ri\ f it will be sufficient to find the coefficients of a:*, y", z^ in 
the function on the left-hand side of the equation. The coefficient of a^ in 11 is 

{a;,^ + 2lyiZi) (irj" + 2lz^^) (a^* + 2Za?,ys) 

-^ •Crj •C/2 «*^ 

+ 2? {onxx^y^z + &c.) 
+ 4i'» (x^y^^^t + &c.) 

• +8Z» yiy^i;siz^t\ 

and it is easy to see that representing the function 

(1, 1, 1, /$)8?- 717, 7f - a?, ai7 - fi^f 
by 

(a, b, c, f, g, h, i, j, k, l$a, /8, 7)», 

the symmetrical functions can be expressed in terms of the quantities a, b, &c., and 
that the preceding value of the coefficient of a^ in H is 

a' 
-f 21 (9hj - 6al) 
+ 4Z» (6gk - 3fj - Shi + 31») 
+ 8Z' be; 

and substituting for a, &c. their values, this becomes 

+ 4^' {- 6 (?r + 21^) (pi, + 2Zf C) 

+ 3(i,{:» + 2f?p)((7f + 2i5i;«) 

+ 3 (?.;' + 2?i,r') (»?•?+ 2^P)) 
and reducing, we obtain for the coefiScient of a* in 11 the following expression, 

-18/ ^*^ 

- 24? (p + 1;" + {:») fi^r 

-24Z»(i7»i;»+C»|» + f.7») 



408 



A M£MOIR ON CURVES OF THE THIRD ORDER. 



[146 



Now the coefficient of a^ in F , U is simply F, which is equal to 

f«+ ^«+ f»- 2i;»?»- 2f*f - 2f i7» 

and subtracting, the coefficient ofa^ in F. U —Tl is 

f - 2f i;» - 2f ?» 

- 48^f^^?», 



which is equal to 



(1 + 8P) f (f* - 2fi;» - 2f ?» - 6lvV)' 



The expression last written down is therefore the value of pf, or dividing by ^ we 
have f, and then the values of rj', f are of course known, and we obtain the 
identical equation 



F. U^U = 



(l + 8P)(f^+W+?^)' ^ 



+ (17* 
+ (?* 



2^7» - 2^f» 
2i7?»-2i;p 
2rr - 2?i;» 



6zr-r)y 



and the second factor equated to zero is the equation of the satellite line of 
fa? + i;y + fg: = 0. 

33. The point of intersection of the line f a? + lyy + f-^ = with the satellite line 
^'x + i;'y + f -^ = is the satellite point of the former line ; and the coordinates of the 
satellite point are at once found to be 

: (P-V)(f^ + 2in- 



34. If the primary line ^x + rfy + fy = is a tangent to the cubic, then (a?,, y,, ^i) 
being the coordinates of the point of contact, we have 

^ : 1) ■ ?=«i' + 2/yi^, : yi*-\r'ilz^ : «i*+2^,; 



146] 



A MEMOIR ON CURVES OF THE THIRD ORDER. 



409 



these values give as before 

and they give also 

and consequently we obtain 

that is, the satellite point of a tangent of the cubic is the point in which this 
tangent again meets the cubic. 



Article Nos. 35 and 36. — Theorems relating to the satellite point. 

35. If the line f a? 4- i;y + ?2r = be a tangent of the Pippian, then the locus of 
the satellite point is the Hessian. 

Take (x, y, z) as the coordinates of Jbhe satellite point, then we have 

x:y : ^ = (i;» - ?») (17?+ 2/p) 
:(C-P)(Sf+2Zi;») 

where the parameters f, 1;, 1^ are connected by the equation 

-i(?+i;'+C')+(-i+4P)fi;r=0. 

We have 

&ad it is easy to see that the function on the right-hand side must divide by tf — i^: 
hence a^ + y' + s^ will also divide by 17* — f*, and consequently by (ij* - {?) (S* — ?) (? — «?*)• 
We have 

I + ? 
+ 6?fV?* { - ^ - rV - 1/' + 3p (?» + 1;») - 3f ) 
+ 12?^,? {-rf^(rf+^) + 3p,^C - f } 

+ 8i» { - i/'f + 3i;»r? - (V + D PI 



and 



C. II. 



52 



410 A MEMOIR ON CURVES OF THE THIRD ORDER. [146 

Adding these values and completing the reduction, we find 

(*■» + y* + z») ^ (»?'- f) (?'-?) (f» - »7») = - ^ - 1?« -?•+ Zi/T + 2{:»? + 2f»7' 

+ 18i ^*^ 
+ 12P(|' + i7' + f»)fi,? 
+ 8i» (i/'C + Cf + fi?») ; 

and we have also 

+ 8Z' f«v'r, 
and thence 

-A i^ + ri' + ^y 

+ (UPA + ilB) (f* + 17» + f») fi^r 

+ (18^4 + (1 + 8?) 5) pi,«i;» 

+ ((4P + 81*) A + 4P5) (vT + rf + f 'j')- 

The coefficient of ij«?» + f »f» + f <;» on the right-hand side will vanish if (1 + 2?) -4 + i* 5 = 0, 
or, what is the same thing, if A=l^, 5 = — (1 + 2Z'); and substituting these values, we 
obtain 

{P {x> + y' + z>)-(l + 2P) f,C} H- (i;* - ?») (C - p) (p - ,;») 

= -P (f + 'T' + H 

+ (- 4Z + 4i*) (t' + 1?' + f) fijf 
+ (- 1 + 8P - lei*) f*ij»?», 

or, what is the same thing, 

P (a;* + y" + ^) - (1 + 2P) a;y^ = - (i,» - H (r - f ) (? -• 7*) 

x[-li^ + v'+^) + (-l + U*) f.7?}'. 

Hence the left-hand side vanishes in virtue of the relation between ^, rj, ^, or we have 

^{x> + f-i-z')-{l + 2P)xyz = 0. 
which proves the theorem. 

36. Suppose that {X, Y, Z) are the coordinates of a point of the Hessian, and 
let (P, Q, R) be the coordinates of the point in which the tangent to the Hessian 
at the point {X, Y, Z) again meets the Hessian, or, what is the same thing, the 



146] 



A MEMOIB ON CURVES OF THE THIRD ORDER. 



411 



satellite point in regard to the Hessian of the tangent at (X, F, Z), And consider 
the conic 

X{x'^' 2lyz) + Tif-^^ 2lzx) + Z(a^'{- 2lxy\ 

which is the first or conic polar of the point (X, F, Z) in respect of the cubic. The 
polar (in respect to this conic) of the point (P, Q, R) will be 



w 



here 



f = PZ + /(i2F+(2Z), 

i=RZ + l(QX+PY); 

or putting for (P, Q, iJ) their values, 

^ = (Y''-Z')(X*-IYZ), 
V = (Z'-X')(7^^1ZX), 
r = (Z»-P)(^-iZF); 

and if from these equations and the equation of the Hessian we eliminate {X, F, Z) 
we shall obtain the equation in line coordinates of the curve which is the envelope 
of the line fa? + i;y + 5^ = 0. We find, in fact, 

p + ,7»+ r = (F« - iP'XZ'- Z») (Z» - F«) 

SI {X'+T' + Z*)XYZ 
+ 91* X^Y^Z^ 
(,+ (!- W){Y^Z* + ^Z» + Z^F'), 



X -^ 



^? 



= (F'-^)(^ - X»)(X»- P) 



r 



J 



Z'(-y*+F' + ^)ZF^ 



i (7'^ + ^Z» + Z»F'); 

and thence recollecting that 

HU^J? (Z» + F» + -Z») - (1 + 2P) ZF-^, 
we find 

and the equation of the envelope is 

which is therefore the Pippian. We have thus the theorem: 



52—2 



412 A MEMOIR ON CURVES OF THE THIRD ORDER. [146 

The envelope of the polar of the satellite point in respect to the Heesian of the 
tangent at any point of the Hessian, such polar being in respect of the conic which 
is the first or conic polar of the point of the Hessian in respect of the cubic, is the 
Pippian. 



Article Nos. 37 to 40. — Investigations and theorems relating to the first or conic polar 

of a point of the cubic. 

37. The investigations next following depend on the identical equations 
[a(X^ + 2lYZ)-{-l3{Y'-^2lZX)-^y(Z^ + 2lXY)] 

= {Z (iB* + 2lyz) + F (y« + 2lzx) + Z (-«» + 2%)} 

-h {x (Z« + 21YZ) + y (F» + 21ZX) '{-z{Z'-\- 21XY)\ 

x{-(aF^-fi9ZZ-f7ZF)(Za;»+Fy» + Z^«) + (aZ« + i9F» + 7Z*)(ZF?+Fzx + Za;y)l. 

which is easily verified. 

I represent the equation in question by 

then considering (a?, y, z) as current coordinates, and (Z, Y, Z) and (a, /8, 7) as the 
coordinates of two given points 2 and fl, we shall have 17=0 the equation of the 
cubic, TT = the equation of the first or conic polar of S with respect to the cubic, 
P = the equation of the second or line polar of 2 with respect to the cubic The 
equation T = is that of a syzygetic cubic passing through the point 2 : the 
coordinates of the satellite point in respect to this syzygetic cubic of its tangent at 
2 are 

XiY^'-Z') : Y{Z^^X^) : Z(Z»- F»); 

and calling the point in question 2^ then Z = is the equation of a line through 
the points 2', ft. The equation = is that of a conic, viz. the first or conic polar 
of 2 with respect to a certain sjrzygetic cubic 

- 2 {aYZ-^^ZX + 7ZF) («» + y» + ^) + (aZ» + /8F» + 7^)ajyir = 0, 

depending on the points 2, ft, or, what is the same thing, the conic B = is a 

properly selected conic passing through the points of intersection of the first or conic 

polars of 2 with respect to any two sjrzygetic cubics; and lastly, .E' is a constiint 
coefficient. The equation expresses that the points of intersection of 

(ir=o, p = o), (Tr=o, e = o), (Z = o, p=o), (Z = o, e = o), 

lie in the syzygetic cubic T = 0. 



146] A MEMOIB ON CURVES OF THE THIRD OKDEB. 413 

The left-hand side of the equation may be written 

- ZF^ {a (Z« + 2ZFZ) + /8 ( F« + 2/ZZ) + 7 (^ + 2iZF)l(rc» + y« + ^ + eZfljy^) 

+ xyz{a(X^ + 2l7Z)+fi(7^ + 2lZX) + y{Z» + 2lXY)}(X^-^Y^ + Z' + 6lXYZ); 

and it may be remarked also that we have 

-3ZFZ{a(Z» + 2iFZ) + /9(F« + 2ZZZ) + 7(^ + 2/ZF)} 

equal identically to 

(Z(F»-Z»)(7F-/8^)+F(Z*-.Z»)(a^-7Z) + ^(Z«-F»)(i8Z-aF)} 

-(aFZ + /SZZ + 7ZF)(Z«+ Y' -^ Z' + eiXYZ). 

Hence if we assume 

Z»+ F» + Z» + 6?ZFZ=0, 
the equation will take the form 

where the constant coefficient K may be expressed under the two different forms 

if = -ZFZ{a(Z« + 2iFZ) + /8(F» + 2/ZZ) + 7(^ + 2ZZF)} 

= J {Z(F' - Z*) (7F- i8Z)+ F(^- Z«)(aZ- 7Z) + Z(Z»- Y*){^X - aF)}, 

and W, Z, P, have the same values as before. In the present case the point S 
is a point of the cubic : the equation IT = represents the first or conic polar of 
the point in question, and the equation P = its second or line polar, which is also 
the tangent of the cubic. The line Z = is a line joining the point XI with the 
satellite point of the tangent at S, or dropping altogether the consideration of the 
point ft, is an arbitrary line through the satellite point: the first or conic polar of 
2 meets the cubic twice in the point 2, and therefore also meets it in four other 
points; the conic © = is a conic passing through these four points, and com- 
pletely determined when the particular position of the line through the satellite 
point is given. And, as before remarked, = is a conic passing through the points 
of intersection of the first or conic polars of 2 with respect to any two syzygetic 
cubics. We have thus the theorem : 

The first or conic polar of a point of the cubic touches the cubic at this point, 
and besides meets it in four other points; the four points in question are the points 
in which the first or conic polar of the given point in respect of the cubic is 
iutersected by the first or conic polar of the same point in respect to any syzygetic 
cubic whatever. 

38. The analytical result may be thus stated: putting 

K = ^YZ'^/3ZX + yXY, X = aZ« + /SF» + 7Z«, 



414 



A MEMOm ON CUBVES OF THE THULD ORDER. 



[146 



or, if we please, considering k, \ as arbitrary parameters, then the four points lie in 
the conic 

(2kX, 2/cT, 2/cZ, -\Z, -\F, -XZ$a?, y, 2r)> = 0, 

or, what is the same thing, they are the points of intersection of the two conies 

Za^ + Fy« + Zz^ = 0, 
Xyz + Yzx + Zxy = 0. 

39. Considering the four points as the angles of a quadrangle, it may be shown 
that the three centres of the quadrangle lie on the cubic. To effect this, assume 
that the conic 

(2/cZ, 2k7, 2kZ, -XZ, -\F, -\Z\x, y, z>^=^0 

represents a pair of lines; these lines will intersect in a point, which is one of the 
three centres in question. And taking x, y, z a& the coordinates of this point, we 
have 

a^ : y^ \ z^ : yz I zx I xy ^ 4/^* YZ — X'Z* 

4/e»ZZ- X»F» 
4/e» XT" X»^ 

X» YZ + 2k\X^ 

X»ZZ + 2/eXF« 

X«ZF+2/eX^; 

and we may, if we please, use these equations to find the relation between /c, X 
Thus in the identical equation a^ . y* — (fl?y)" = 0, substituting for a^, xy, y" their values, 
and throwing out the factor Z, we find (4#c>-X»)ZFZ-/eX«(Z«+ F« + ^) = 0, and 
thence, in virtue of the equation Z'+ F* + ^ + 6?ZFZ = 0, we obtain 

4#c> -f- 6?/eX» - X» = 0. 

But the preceding system gives conversely, 

X* : Y* : Z" : YZ : ZX : XY^^i^z- XV 

^i^zx - xy 

4/c'a?y — XV 
Xh/z + 2/cXflj« 
X^zx + 2/cXy* 
X^xy + 2/eX«». 

Hence firom the identical relation Z*. F»-(ZF)» = 0, substituting for X\ XY, P 
their values, and throwing out the £sbctor z, we find (4ie* — X')a5y2r — /eX*(«* + y' + z*) = 0, 
and thence, in virtue of the equation 4#c* — X' = — 6i/icX*, we obtain 

^ + y* + ^ + Qlxyz = 0, 



146] A MEMOIB ON CURVES OF THE THIRD ORDER. 415 

which shows that the point in question lies on the cubic. We have thus the 
theorem : 

The first or conic polar of a point of the cubic touches the cubic at the point, 
and meets it besides in four points, which are the angles of a quadrangle the 
centres of which lie on the cubic. In other words, the quadrangle is an inscribed 
quadrangle. 

40. To find the equations of the three axes of the quadrangle, that is of the 
lines through two centres. 

We have 

(4/e» 7Z - \^X') x + ( \'XY-^ ^KkZ") y + ( X»^Z + 2/eX7«) ^ = 0, 
( \^XY-{- 2k\Z^) X + (4^ZX - X«r») y + ( X'FZ + 2ic\Z«) ^ = 0, 
( X'ZZ + 2/rXF>) a? + ( X«7Z + 2/rXZ»)y+(4/e»ZF- X«Z0-r=O; 

or arranging these equations in the proper form and eliminating ic", k\ X', we find 

YZx, Z^-\-Y^z, X{-Xx-^Yy-\-Zz) =0; 
ZXy, XH^-Z'x , Y{ Xx-Yy-^- Zz) 
XYz, r»a? + Z«y, Z{ Xx -\- Yy -\- Zz) 

or, multiplying out, 

ZFZ{(^^F»)a;» + (Z»-^)y» + (r»-Z»)-»'} 

-f a^yZY^ (- 2Z» + F» + ^) + zai'YZ^ (2X^ --Y^^Z*) 
+ y'zXZ^ (- 2 F» + ^ -f Z») + xfZX* (2F» - 2> - Z») 
+ z'xYX*(- 2Z* + Z» + F») + y^»ZF'(2^ - Z* - F») = 0. 

We may simplify this result by means of the equation Z»+ F» + ^ + 6ZZFZ= 0, so as 
to make the left-hand side divide out by XYZ: we thus obtain 

(^-F»)a:» + (Z»-i?')y» + (F»-Z»)-?' 
+ (- 3Z»F-. 6iF>Z)^ + (- 3F»Z- 6lZ'X)y*z + (- 3^Z - 6lX'Y)^x 

+ ( 3ZF«+6ZZ>Z)«y« + ( 3FZ*+6/F»Z)y^»-|-( 3^Z^ + 6/^F)^a^ = (); 
or in a difierent form, 

+ (- Sx'y - eiz^x) X^Y+(- 3y'z - Gla^^y) Y'Z + (- 3z'x - 6ly^z) Z'X 

+ ( fixy''h6lyz^)XY^ + ( Syz^ + 6lzaf) YZ' + ( Szx' -\- Qlxf) ZX^ == 0, 

as the equation of the three axes of the quadrangle. 



416 A MEMOIR ON CURVES OF THE THIRD ORDER. [l46 



Article No. 41. RecapitulcUum of geometrical definxtione of the Pippian. 

In conclusion, I will recapitulate the different modes of generation or geometrical 
definitions of the Pippian, obtained in the course of the present memoir. The curve 
in question is: 

1. The envelope of the line joining a pair of conjugate poles of the cubic (see 
Nos. 2 and 13). 

2. The envelope of each line of the pair forming the first or conic polar with 
respect to the cubic of a conjugate pole of the cubic (see Nos. 2 and 14). 

3. The envelope of a line which is the polar of a conjugate pole of the cubic, 
with respect to the conic which is the first or conic polar of the other conjugate pole 
in respect to any syzygetic cubic (see Noa 2 and 9). 

4. The locus of the harmonic with respect to a pair of conjugate poles of the 
cubic of the third point of intersection with the Hessian of the line joining the two 
conjugate poles (see Nos. 2 and 17). 

6. The envelope of a line such that its lineo-polar envelope with respect to the 
cubic breaks up into a pair of lines (see No. 24). 

6. The envelope of a line which meets three conies, the first or conic polars of 
any three points in respect to the cubic, in six points in involution (see No. 22^ 

7. The envelope of the second or line polar with respect to the cubic, of a point 
the locus of which is a certain curve of the sixth order in quadratic syzygy with 
the cubic and Hessian, viz. the curve — S . IT"* + (fiVy = (see No. 27). 

8. The envelope of a line having for its satellite point a point of the Hessian 
(see No. 35). 

9. The envelope of the polar of the satellite point with respect to the Hessian 
of the tangent at a point of the Hessian, with respect to the first or conic polar of 
the point of the Hessian in respect to the cubic (see Na 86). 



147. 



I 
I 



A MEMOIR ON THE SYMMETRIC FUNCTIONS OF THE ROOTS 

OF AN EQUATION. 

[From the Philosophical Transactions of the Royal Society of London, vol. CXLVII. for 
the year 1857, pp. 489 — 499. Received December 18, 1856, — Read January 8, 1857.] 

There are contained in a work, which is not, I think, so generally known as it 
deserves to be, the "Algebra" of Meyer Hirsch [the work referred to is entitled 
Sammlung von Beispielen Formeln und Aufgaben aus der Buchstabenrechnung und 
Algebra, 8vo. Berlin, 1804 (8 ed. 1853), English translation by Ross, 8vo. London, 
1827] some very useful tables of the symmetric functions up to the tenth degree 
of the roots of an equation of any order. It seems desirable to join to these a set of 
tables, giving reciprocally the expressions of the powers and products of the coefficients 
in terms of the symmetric functions of the roots. The present memoir contains the 
two sets of tables, viz. the new tables distinguished by the letter (a), and the tables 
of Meyer Hirsch distinguished by the letter (6) ; the memoir contains also some 
remarks as to the mode of calculation of the new tables, and also as to a peculiar 
symmetiy of the numbers in the tables of each set, a synmietry which, so far as I 
am aware, has not hitherto been observed, and the existence of which appears to 
constitute an important theorem in the subject. The theorem in question might, I 
think, be deduced from a very elegant formula of M. Borchardt (referred to in the 
sequel), which gives the generating function of any sjonmetric function of the roots, 
and contains potentially a method for the calculation of the Tables (6), but which, 
fix)m the example I have given, would not appear to be a very convenient one for 
actual calculation. 

Suppose in general 

(1, 6, C...5I, a?)* =(1 — 6a?)(l— )8a:)(l — 7a:)... , 
so that 

— 6 = 2a, + c = 2a)8, — d = %afiy, &c., 
and if in general 

0. n. 53 



418 



A MEMOIB ON THE SYMMETRIC FUNCTIONS 



[147 



where as usual the summation extends only to the distinct terms, so that e.g. (/?*) 
contains only half as many terms as (pq), and so in all similar cases, then we have 

-6 = (l), +c = (P), -d = (P), &c.; 

and the two problems which arise are, first to express any combination fr'^c^... in terms 
of the symmetric functions (l^m^..,), and secondly, or conversely, to express any 
symmetric function {l^m^ ...) in terms of the combinations b^d^,... 

It will conduce materially to brevity if 1^29... be termed the partition belonging 
to the combination b^d^... ; and in like manner if l^m^... be termed the partition 
belonging to the symmetric function (l*m^,,,), and if the sum of the component 
numbers of the partition is termed the weight. 

Consider now a line of combinations corresponding to a given weight, e.g. the 
weight 4, this will be 

e bd d" b^c b" (line) 
4 13 2» 1«2 1*, 

where I have written under each combination the partition which belongs to it, and 
in like manner a colunm of symmetric functions of the same weight, viz. 

(4) (column) 

(31) 

(2») 

(21*) 
(1*). 

where, as the partitions are obtained by simply omitting the ( ), I have not separately 
written down the partitions. 

It is at once obvious that the different combinations of the line will be made up 
of numerical multiples of the symmetric functions of the column ; and conversely, that 
the symmetric functions of the column will be made up of numerical multiples of the 
combinations of ^ the line ; but this requires a further examination. There are certain 
restrictions as to the symmetric functions which enter into the expression of the com- 
bination, and conversely, as to the combinations which enter into the expression of the 
symmetric function. The nature of the first restriction is most clearly seen by the 
following Table: 



Number of 


Greatest 


Parts. 


Part. 


1 


4 


2 


3 


2 


2 


3 


2 


4 


1 



Combinations 

with their several 

Partitions. 



6 
bd 

b^o 
b* 



4 

13 

2« 

1«2 

1* 



Contain Maltiples of the 
Symmetrio Funotions. 



(1*), 



(21»), 
(21'), 



(2'), 

(2^), (31), 

(2'), (31), (4) 



Greatest Part 
does not exceed 



1 
2 
2 
3 
4 



Number of j 
Parts not 
less than 



4 
3 
2 
2 
1 



147] OF THE ROOTS OF AN EQUATION. 419 

Thus, for instance, the combination bd (the partition whereof is 13) contains multiples 
of the two symmetric functions (1*), (21*) only. The number of parts in the partition 
13 is 2, and the greatest part is 3. And in the partitions (1*), (21') the greatest part 
is 2, and the number of parts is not less than 3. The reason is obvious : each term of 
the developed expression of bd must contain at least as many roots as are contained 
in each term of d, that is 3 roots, and since the coefficients are linear functions in 
respect to each root, the combination bd cannot contain a power higher than 2 of any 
root. The reasoning is immediately applied to any other case, and we obtain 

First Restriction. — A combination b^c^... contains only those symmetric functions 
(l*mv.,,), for which the greatest part does not exceed the number of parts in the 
partition P2^... , and the number of parts is not less than the greatest part in the 
same partition. 

Consider a partition such as 1*2, then replacing each number by a line of units 
thus, 

1 

1 

11, 

and summing the columns, we obtain a new partition 31, which may be called the 
conjugate^ of 1*2. It is easy to see that the expression for the combination 6*c (for 
which the partition is 1'2) contains with the coefficient unity, the symmetric function 
(31). the partition whereof is the conjugate of 1*2. In fact 6"c = (— 2a)" (Sa/8), which 
obviously contains the term + lo*^, and therefore the symmetric function with its 
coefficient + 1 (31) ; and the reasoning is general, or 

Theorem. A combination 6^c^... contains the sjrmmetric function (partition conjugate 
to 1''2^...) with the coefficient unity, and sign + or — according as the weight is even 
or odd. 

Imagine the partitions arranged as in the preceding column, viz. first the partition 
into one part, then the partitions into two parts, then the partitions into three parts, 
and so on; the partitions into the same number of parts being arranged according to 
the magnitude of the greatest part (the greatest magnitude first), and in case of 
equality according to the magnitudes of the next greatest part, and so on (for other 
examples, see the outside column of any one of the Tables). The order being thus 
completely defined, we may speak of a partition as being prior or posterior to another. 
We are now able to state a- second restriction as follows. 

Second Restriction. — The combination b^cfi.,. contains only those symmetric functions 
which are of the form (partition not prior to the conjugate of 1*2*...). 

The terms excluded by the two restrictions are many of them the same, and it 
might at first sight appear as if the two restrictions were identical; but this is not 

^ The notion of Conjugate Partitions is, I believe, due to Professor Sylvester or Bfr Ferrers. [It was dne to 
Mr now Dr Ferrers.] 

53—2 



420 



A MEMOIR ON THE SYMMETRIC FUNCTIONS 



[147 



so : for instance, for the combination bd\ see Table VII (a), the term (41*) is excluded 
by the first restriction, but not by the second; and on the other hand, the term 
(3*1), which is not excluded by the first restriction, is excluded by the second restriction, 
as containing a partition 3^1 prior in order to 32', which is the partition conjugate 
to 13', the partition of bd\ It is easy to see why b(P does not contain the symmetric 
function (3'1); in fact, a term of (3*1) is (a'/3'7), which is obviously not a term of 
6d' = (— 2a) (Sa^y)* ; but I have not investigated the general proof 

I proceed to explain the construction of the Tables (a). The outside column 
contains the symmetric fimctions arranged in the order before explained; the outside or 
top line contains the combinations of the same weight arranged as follows, viz. the 
partitions taken in order from right to left are respectively conjugate to the partitions 
in the outside column, taken in order from top to bottom ; ia other words, each square 
of the sinister diagonal corresponds to two partitions which are conjugate to each other. 
It is to be noticed that the combinations taken in order, from left to right, are not 
in the order in which they would be obtained by Arbogast's Method of Derivations 
from an operand a*, a being ultimately replaced by unity. The squares above the 
sinister diagonal are empty (i.e. the coefficients are zero), the greater part of them in 
virtue of both restrictions, and the remainder in virtue of the second restriction; the 
empty squares below the sinister diagonal are empty in virtue of the second restriction; 
but the property was not assumed in the calculation. 

The greater part of the numbers in the Tables (a) were calculated, those of each 
table fi"om the numbers in the next preceding table by the following method, 
depending on the derivation of the expression for b^^^c^... from the expression for b'^c^... 
Suppose, for example, the column cd of Table V(a) is known, and we wish to calculate 
the column bed of Table VI (a). The process is as follows : 



Given 



we obtain 



2n 21V 1» 



3 10 



321 


2' 


31» 


2n« 


21* 


1« 


1 


3 




2 










3 


6 


12 
10 


60 


1 


3 


3 


8 


22 


60 



where the numbers in the last line are the numbers in the column bed of Table 
VI (a). The partition 2*1, considered as containing a part zero, gives, when the parts 
are successively increased by 1, the partitions 321, 2', 2*1', in which the indices of the 
increased part (i.e. the original part plus unity) are 1, 3, 2; these numbers are taken 
as multipliers of the coefficient I of the partition 2'1, and we thus have the new 
coefficients 1, 3, 2 of the partitions 321, 2*, 2*1*. In like manner the coefficient 3 of 



147] OF THE ROOTS OF AN EQUATION. 421 

the partition 21' gives the new coefficients 3, 6, 12 of the partitions 31', 2*1*, 21*, 
and the coefficient 10 of the partition 1* gives the new coefficients 10, 60 of the 
partitions 21* and 1", and finally, the last line is obtained by addition. The process 
in fact amounts to the multiplication separately of each term of cd = 

1 (2»1) + 3 (21') + 10 (1») 

by 6 = (1). It would perhaps have been proper to employ an analogous rule for the 
calculation of the combinations M^.,. not containing 6, but instead of doing so I 
availed myself of the existing Tables (6). But the comparison of the last line of each 
Table (a) (which as corresponding to a combination b^ was always calculated in- 
dependently of the Tables (6)) with such last line as calculated from the corresponding 
Table (6), seems to afford a complete verification of both the Tables ; and my process 
has in fact enabled me to detect several numerical errors in the Tables (6), as given 
in the English translation of the work above referred to. It is not desirable, as 
regards facility of calculation and independently of the want of verification, to calculate 
either set of Tables wholly from the other; the rules for the independent calculation 
of the Tables (6) are fully and clearly explained in the work referred to, and I have 
nothing to add upon this subject. 

The relation of symmetry, alluded to in the introductory paragraph of the present 
memoir, exists in each Table of either set, and is as follows: viz. the number in the 
Table corresponding to any two partitions in the outside column and the outside line 
respectively, is equal to the number corresponding to the same two partitions in the 
outside line and the outside column respectively. Or, calling the two partitions P, Q, 
and writing for shortness, combination (P) for the combination represented by the 
partition P, and for greater clearness, symmetric function (P) (instead of merely (P)) 
to denote the symmetric function represented by the partition P, we have the following 
two theorems, viz. 

Theorem. The coefficient in combination (P) of symmetric function (Q) is equal 
to the coefficient in combination (Q) of symmetric function (P); 

and conversely. 

Theorem. The coefficient in symmetric function (P) of combination (Q) is equal 
to the coefficient in symmetric function (Q) of combination (P). 

M. Borchardt's formula, before referred to, is given in the *Monatsbericht' of the 
Berlin Academy (March 5, 1885) S and may be thus stated; viz. considering the case of 
n roots, write 

(1, 6, c, ... k"$l, xf = (1 - aa;)(l - ^a?)...(l - kx) =/b, 

then 

1 1 _i ^N ^ 1 / x„ My-'-f^ A ^ A. n(^,y,...ti) 

J -ax l--^y'"l- Ku) k^ ^ Il(Xyy,...u) dx dy" du fxjy...fu ' 

1 And in Crelle, t. liii. p. 195.— Note added 4th Deo. 1857, A. C. 



422 A MEMOIR ON THE SYMMETRIC FUNCTIONS [147 

where 11 (x, y,.,,u) denotes the product of the differences of the quantities x, y,..,v, 
and on the left-hand side the summation extends to all the different permutations of 
a, ^, ... /t, or what is the same thing, of x, y,...u. 

Suppose for a moment that there are only two roots, so that 

(1, 6, cth xy^(l^ax)(l^l3xl 

then the left-hand side is 

1 1 

which is equal to 

2 + (a + i9)(a? + y) + (a» + ^)(a;» + y') + 2a/8ajy + (a» + /80(aJ» + y') + (a»i8 + <^^ 

and the right-hand side is 

1 fx/y d d x-y 



which is equal to 



c ' x-y dx dy fxfy ' 



1 My [ r^fy-ryfaH^-y)rxfy \ 



c x — y 



(f^r {fyf 



and therefore to 



c fxfy { x-y J J i^y 



or substituting for fx^ fy their values, 

f^fy -fyfa 

x-y 
becomes equal to 

2c - 62 - 6c(a; + y) - 2c»icy, 
and fxfy is equal to 

6» + 2ic (a? H- y) + ^xy. 

The right-hand side is therefore equal to 

2 + 6(a? + y) + 2cay . 

(l+6a? + cic»)(l + 6y + cy«)' 

and comparing with the value of the left-hand side, we see that this expression may 
be considered as the generating function of the sjrmmetric functions of (a, /S), viz. the 
expression in question is developable in a series of the symmetric functions of (a:, y\ 
the coefficients being of course functions of 6 and c, and these coefficients are (to 
given numerical factors yrls) the symmetric functions of the roots (a, /8). 



147] 



OP THE ROOTS OP AN EQUATION. 



423 



And in general it is easy to see that the left-hand side of M. Borchardt's 
formula is equal to 

[0] + [1] (1) (ly + [2] (2) (2)' + [P] (1») (1')' + &c, 

where (1), (2), (1'), &c. are the symmetric functions of the roots (a, A ••• fc)y (1)', (2)', 
{V)\ &c. are the corresponding symmetric functions of («, y, ...w), and [0], [1], [2], [1*], 
&c. are mere numerical coefficients; viz. [0] is equal to 1.2.3...n, and [1], [2], [1'], &c. 
are such that the product of one of these factors into the number of terms in the 
corresponding sjrmmetric function (1), (2), (P), &c. may be equal to 1.2.3...n. The 
right-hand side of M. Borchardt's formula is therefore, as in the particular case, the 
generating function of the symmetric functions of the roots (a, ^, ... ^), and if a 
convenient expression of such right-hand side could be obtained, we might by means 
of it express all the sjrmmetric functions of the roots in terms of the coefficienta 



Tahles rdating to the Symmetric Functions of the Roots of an EquaMon. 

The outside line of letters contains the combinations (powers and products) of the 
coefficients, the coefficients being all with the positive sign, and the coefficient of the 
highest power being unity; thus in the case of a cubic equation the equation is 

a;^ + 6«* + w? + d = (a? - a) (a? - )8) (a? - 7) = 0. 

The outside line of numbers is obtained from that of letters merely by writing 1, 2, 3... 
for 6, c, d..., and may be considered simply as a different notation for the combinations. 
The outside column contains the different symmetric functions in the notation above 
explained, viz. (1) denotes So, (2) denotes So", (1') denotes SayS, and so on. The Tables 
(a) are to be read according to the columns; thus Table 11(a) means 6" = 1 (2) + 2(1)*, 
c = (l*). The Tables (6) are to be read according to the lines; thus Table 11(6) 
means (2) = - 2c + 1ft*, (1») = + Ic. 



1(a). 



(1)1-1 



1(6). 



(1) 



1 

6 

- 1 



II (a). 



(2) 
(I'O 



2 
c 

+ 1 


1» 

+ 1 

+ 2 



II (6). 



2 
c 



P 
6* 



(2) 
(1') 






m(o). 





3 


12 


1» 


11 


d 


he 


6» 


(3) 






- 1 


(21) 




- 1 


-3 


(1') 


-1 


-3 


-6 




m (6). 




3 12 


1» 




d he 


6» 


(3) 


-31 + 3 


- 1 


(21) 


+ 3-1 




(1») 


- 1 





424 



A MEMOIR ON THE SYMMETRIC FUNCTIONS 



[147 



IV (a). 



(5) 
(41) 

(32) 
(31«) 

(2'1) 

(2P) 

(!•) 



II 

(4) 
(31) 

(2«) 

(21') 

(1^) 



4 


13 


2^ 


P2 


V 


e 


bd 


c* 


b^c 


b* 










+ 1 






+ 1 


+ 1 

+ 2 


+ 4 


+ 6 


+ 1 


+ 1 
+ 4 


+ 2 
+ 6 


+ 5 


+ 12 


+ 12 


+ 24 



V(o). 



5 
/ 

- i 


14 
be 

- 1 
-5 


23 
cd 


P3 
b*d 


12* 
6c« 


P2 
b^c 


b' 






1 
- 3 


1 








- 5 




1 
- 2 


- 10 




- 1 


- 7 

- 12 
-27 


- 20 


- 1 


- 2 

- 7 


- 5 


- 30 

- 60 


- 3 
10 


-12 


-20 


-30 


-60!- 120 







IV 


(b). 








i 13 


2« 


1«2 


I* 


^ 


« 


bd 


c« 


ft»c 


6* 


(4) 


-4 


+ 4 


+ 2 


- 4 


+ 1 


(31) 


+ 4 


- 1 


-2 


+ 1 




(2') 


+ 2 


-2 


+ 1 






(21') 


-4 


+ 1 








(1') 


+ 1 

















V 


(6). 










5 


14 


23 


P3 


12« 


P2 


P 


= 


/ 


be 


cd 


b^d 


b<^ 


6»c 


6» 


(5) 


-5 


+ 5 


+ 5 


-5 


5 


+ 5 


-1 


(41) 


+ 5 


- 1 


-5 


+ 1 


+ 3 


- 1 




(32) 


+ 5 


-5 


+ 1 


+ 2 


- 1 






(31») 


-5 


+ 1 


+ 2 


- 1 








(2*1) 


- 5 


+ 3 


- 1 










(2P) 


+ 5 


- 1 












(P) 


- 1 










1 



II 


6 
9 


15 

¥ 

+ I 


24 
ce 


P4 
b^e 


3» 
d" 


VI (a] 

123 
bed 


1. 

P3 
b^d 


2» 


P2« 


P2 
b'c 


6« 


(6) 












i 


+ 1 


(51) 
















+ 1 


+ 6 


(42) 












+ l!+ 4 


+ 15 


(3') 













+ 1 


+ 2 


+ 6 


+ 20 


(41') 








+ 1 


• • • 


+ 2 


+ 9 


+ 30 


(321) 






+ 1 


+ 3 


+ 3 
+ 6 


+ 8 


+ 22 


+ 60 


(2') 






+ 1 


+ 3 


+ 6 


+ 15 


+ 36 


+ 90 


(31') 




+ 1 


• • • 


+ 3 


+ 10 


+ 6 


+ 18 


+ 48 


+ 120 


(2»1') 


+ 1 


+ 2 


+ 2 


+ 8 


+ 18 


+ 15 


+ 34 


+ 78 


+ 180 


(21*) 


+ 4 
+ 15 


+ 9 


+ 6 


+ 22 


+ 48 


+ 36 


+ 78 


+ 168 


+ 360 


(1-) 


+ 1 


+ 6 


+ 30 


+ 20 


+ 60 


+ 120 


+ 90 


+ 180 


+ 360 


+ 720 



VI (6). 



(6 
(51 

(42 

(3« 

(4P 

(321 

(2" 

(31 

(2«P 

(2P 



6 
9 


15 
¥ 
+ 6 


24 
ee 

+ 6 


P4 
6»c 

-6 

+ 1 

+ 2 

+ 3 

- 1 

-3 

• • • 

+ 1 


3« 
d* 

+ 3 

-3 

-3 

+ 3 

+ 3 

-3 

+ 1 


123 
bed 


P3 
b^d 

+ 6 

- 1 

- 2 

• • • 

+ 1 


2» 
c» 

- 2 

+ 2 

2 

+ 1 


P2> 
6V 

+ 9 

-4 

+ 1 


P2 
b*c 

-6 

+ 1 


1* 
+ 1 


- 6 


- 12 


+ 6 


-l|-6 


+ 7 


+ 6 


-6 
-3 
+ 1 

+ 7 
+ 2 
- 1 
-4 
+ 1 


+ 2 
-3 
+ 2 
+ 4 
-2 
2 
+ 1 


+ 4 


+ 3 


- 3 


- 6 


- 3 


- 12 


+ 1 


- 2 




+ 6 




+ 9 
- 6 






+ 1 











147] 



OP THE ROOTS OF AN EQUATION. 



425 



7 
h 


16 
^9 


25 


1«5 


34 
de 


124 
hce 


1»4 
6»« 


VII ( 

12« 


[a), 

2«3 
c^d 


P23 
h^cd 


P3 
h^d 


12» 
6c» 


P2« 
6V 


P2 
6»c 


V 




- 1 
-7 
























1 


























1 


7 
- 21 
























1 


5 






















- 1 


3 


- 10 


- 35 

- 42 




















- 1 


• • • 


2 


- 11 










1 






- 1 


- 4 


- 3 


- 11 


- 35 


- 105 

















- 1 


- 2 


- 6 


- 7 


- 18 


- 50 


- 140 












- 1 


- 2 


- 5 


- 12 


- 12 


- 31 


- 80 


- 210 










1- 1 


• • • 


• • • 


- 3 


- 13 


- 6 


- 24 


- 75 


- 210 










- 1 


- 3 


- 2 


- 5 


- 13 


- 34 


- 27 


- 68 


- 170 


- 420 








- 1 


- 3 


- 6 

- 13 


- 7 


- 12 


- 27 


- 60 


- 51 


- 117 


270 


- 630 






- 1 


• • • 


- 4 


- 6 


12 


- 34 


- 88 


- 60 


- 150 


- 360 


- 840 




- 1 


- 2 


- 3 


- 11 


- 24 


- 18 


- 31 


- 68 


- 150 


-117 


- 25« 


- 570 


- 1260 




- 5 


- 11 


-10 
-35 


- 35 


- 75 


- 50 


- 80 


- 170 


-360 


-270 


- 570 


- 1200 


-2520 


- 1 


-21 


-42 


- 105 


-210 


- 140 


-210 


-420 


-840 


-630 


- 1260 


-2520 


-5040 



(7) 
(61) 

(52) 


7 
h 

- 7 

+ 7 

+ 7 


16 

hg 

+ 7 

- 1 
-7 
-7 
+ 1 
+ 8 
+ 4 
+ 7 

- 1 
-9 
-5 
+ 1 
+ 5 

- 1 


25 

cf 
+ 7 
-7 
+ 3 
-7 
+ 2 
+ 4 
+ 7 
-3 

- 2 
-6 
+ 3 
+ 2 

- 1 


1»5 

bV 

-7 
+ 1 
+ 2 
+ 7 

- 1 
-3 
-4 
-2 
+ 1 
+ 4 

• • • 

- 1 


34 
de 

+ 7 

-7 

-7 

+ 5 

+ 7 

+ 2 

-5 

+ 1 

-3 

+ 3 

- 1 


124 
bee 

- 14 
+ 8 
+ 4 
+ 2 

- 3 

- 8 
+ 1 
+ 2 
+ 3 

- 1 


VII 

1'4 

6»e 

+ 7 

- 1 
-2 
-3 
+ 1 
+ 3 

• • • 

• • • 

- 1 


(6). 

ll« 
bd^ 

-7 

+ 4 

+ 7 

-5 

-4 

+ 1 

+ 2 

- 1 


2«3 
cH 

-7 

+ 7 

-3 

+ 1 

-2 

+ 2 

- 1 

1 


P23 
h'cd 

+ 21 

- 9 

- 6 
+ 3 
+ 4 

- 1 


P3 
b*d 

-7 

+ 1 

+ 2 

• • • 

- 1 


12' 
b(? 

+ 7 

-5 

+ 3 

- 1 


1322 

- 14 
+ 5 

- 1 


1»2 

+ 7 
- 1 


V 
b' 

- 1 


(43) 
(5P) 


+ 7 

- 7 

- 14 

- 7 


(421) 
(3*1) 






1 


(32«) 
(4P) 

(321«) 
(2*1) 
(31^) 

(2n») 
(2P) 

(10 


- 7 
+ 7 
+ 21 
+ 7 

- 7 

- 14 
+ 7 

- 1 


1 


■ 





c. n. 



54 



426 



A MEMOIR ON THE SYMMETRIC FUNCTIONS 



[147 



VIII (a). Runs on infrit. 



(8) 
(71) 

(62) 

(53) 

(4') 
(6P) 

(521) 

(431) 

(42») 

(3''2) 

(51') 

(421«) 

(3'1») 

(32n) 

(2*) 
(41') 
(321') 
(2'1»; 
(31»; 
(2'1«) 

(21') 
(P) 



8 

• 

1 


17 
bh 


26 

eg 


P6 


35 
df 


125 
be/ 


V5 


4^ 


134 
bde 


2«4 


1«24 
b'^ce 


P4 ; 23« 
b*e \ cd* 


















1 














1 1 

1 














: 1 

















1 


• 




— 


























1 
1 


















1 


1 































1 i 

1 

1 











— — 






1 1- 1 




— . 












+ 1 ... 





















+ 1 


+ 4; ... 














+ 1 


+ 2 


+ 6 , + 2 














+ 1 


+ 2 


+ 5 


+ 1214- 5 













+ 1 


+ 4 


+ 6 


+ 12 


+ 241+ 12 










+ 1 


« • • 


• • • 


• • • 


+ 4 


+ 17; ... 








+ 1 


+ 3 


• • • 


. 3 


+ 7!+ 18 


+ 46 + 12 






+ 1 


+ 3 

+ 5 


+ 6 


+ 2 


+ 11 


+ 1«!+ 39 


+ 84+31 




+ 1 


+ 1 

+ 2- 


• • « 


+ 16 
+ 30 


• • • 


+ 10 
+ 32 


+ 20 1 + 55 


+ 140 -r 30 


1 


^ 4 


+ 14 


+ 53 
+ 150 


+ 114 


+ 246 + 80 

1 


+ 1 


+ 1 

+ 8 


+ 6 

+ 28 


+ 13 


+ 15 
+ 56 


+ 51 


+ 108 


+ 20 
+ 70 


!+ 95 


+ 315 


+ 660 + 210 
+ 1680 ' + 560 

4 


+ 56 


+ lt)8 


+ 336 


+ 280 


+ 420 1 + 840 



■| 


1'3» 


i 


12«3 
he'd 


P23 
iy'cd 


1«3 
bH 


2* 
c* 


P2» 
6V 




P2« 
6V 




1«2 
6«c 






(8) 




1 








+ 


1 


(71) 








1 ' 




- 


+ 


I 


+ 


8 


(62) 






1 

1 ; 




+ 
+ 


1 


+ 


6 


+ 


28 


(53) 












+ 1 


4 


+ 


15 


+ 


56 


(4') 








i+ 1 


+ 2 


+ 


6 


+ 


20 


+ 


70 


(61») 








+ 1 

1 


• • • 


+ 


2 


+ 


13 


+ 


56 


(521) 






+ 1 


+ 5 


• • • 


+ 3 


+ 


14 


+ 


51 


+ 


168 


(♦31) 




+ 


1 


+ 3 


+ 10 


+ 4 


+ 11 


+ 


32 


+ 


95 


+ 


280 


(42') 


+ 1 


+ 


2 


+ 7 


+ 20 
+ 30 


+ 6 


+ 18 


+ 


53 


+ 


150 


+ 


420 


(3»2) 


+ 2 


+ 


5 


+ 12 


+ 12+31 


+ 


80 


+ 


210 


+ 


560 


(31') 


• • • 

+ 2 


• • • 


+ 3 


+ 16 


• • • 


+ 6 


+ 


30 


+ 


108 


+ 


336 


(421') 


+ 


5 


+ 18 


+ 55 + 12 


+ 39 


+ 
+ 


114 
172 


+ 


315 


+ 


840 


(3'1') 


+ 4 


+ 


12 


+ 30 


+ 80 


+ 28 


+ ()8 


+ 


440 


+ 


1120 


(32'1) 


+ 12 


+ 


24 


+ 58 


+ 140 


+ 48 


+ 117 


+ 


284 


+ 


690 


+ 


1680 


(2«) 


+ 28 


+ 


48 


+ 108 


+ 240 


+ 90 


+ 204 


+ 


468 


+ 


1080 


+ 


2620 


(iV) 


+ 6 


+ 


12 


+ 46 


+ 140 


+ 24 


+ 84 


+ 


246 


+ 


660 


+ 


1680 


(321') 


4- 30 


+ 


58 


+ 141 
+ 258 


+ 340 + 108 


+ 258 


+ 
+ 


612 
1008 


+ 


1440 


+ 


3360 


(2'!') 


+ 68 


+ 


117 


+ 570 


+ 204 


+ 453 


+ 


2250 


+ 
+ 


5040 
6720 


(31') 


+ 80 
+ 172 


+ 


140 


+ 340 


+ 800 


+ 240 


+ 570 


+ 


1320 


+ 


3000 


(2'1*) 


+ 


284 


+ 612 


+ 1320 


+ 468 


+ 1008 


+ 


2172 


+ 


4680 


+ 


10080 


(21«) 


+ 440 


+ 


690 


+ 1440 


+ 3000 


+ 1080 


+ 2250 


+ 


4680 


+ 


9720 


+ 


20160 


(1») 


+ 1120 


+ 


1680 


+ 3360 


+ 6720 ' + 2520 


+ 5040 


+ 


10080 


+ 


20160 ^ 


40320 



147] 



OF THE ROOTS OF AN EQUATION. 



427 



VIII (6). Runs on iattk. 



(8) 

(71) 

(62) 

(53) 

(4») 

(6P) 

(521) 

(431) 

(42') 

(3'2) 

(51') 

(421») 

(3n«) 

(32*1) 

(2') 

(41^) 

(321*) 

(2»1») 

(31») 

(2'1*) 

(21') 

(1") 



8 

• 


17 
bh 


26 


P6 

- b 
+ 1 
+ 2 
+ 8 
+ 4 

- 1 
-3 

- 9 

- 2 

- 5 
+ 1 
+ 4 
+ 5 
+ 5 

• • • 

- I 
-5 

• • • 

+ 1 


35 
df 

+ 8 
-8 
-8 
+ 7 

4 
+ 8 
+ 1 
+ 1 
+ 8 
-7 
-3 

9 
+ 3 
+ 6 
- 2 
+ 3 
-3 
+ 1 


125 

w 


P5 

^/ 

+ 8 

- 1 

- 2 
-3 

- 4 

+ 1 


4* 
e« 

+ 4 

-4 


134 
hde 


2M 


1«24 


IH 


23« 


- 8 


+ 8 


+ 8 


- 16 
+ 9 


16 


- 8 


+ 21 


-8-8 
+ 1' + 8 
+ 2! + 2 
+ .3| 7 

... ' + 4 
- 1 - 5 
-3+5 

... 1 — 1 
— 2 

... + 1 

1 

1 


+ 8 


- 1 


- 8 


+ 9 


+ 8 
-4 


- 10 


+ 8 


- 8 


+ 4 


+ 4 


-4 

-4 
+ 6 
+ 4 
+ 8 
-8 
- 4 


+ 16 


- 6 


+ 8 


- 8 


- 8 


+ 1 


+ 1 


+ 8 

- 4 
-2 

- 4 

• • • 


- 9 


+ 4 


- 4 


- 4 


+ 8 


- 8 


+ 4 


8 


+ 1 


+ 2 


- 3 


- 9 


+ 4 


-- ]6 
-16 


+ 9 


+ 4 


- 8 


+ 3 


- 10 


+ 11 


+ 9 


+ 16 


10 


+ 4 
+ 2 

• • • 


+ 10 


- 1 


8 
- 8 


+ 8 
+ 8 
- I 


- 4 


4 


• • • 


+ 4 
- 2 
+ 2 
-2 

+ 1 


- 2 

• • • 


+ 2 


+ 5 


+ 4 
- \ 


1 


+ 8 


_ 2 


+ 3 


- 1 

- 4 

■ • • 

• • • 

• • • 

+ 1 


+ 4 


-- 4 


+ 24 


- 10 


- 6 


H- 11 


+ 4 


- 1 


+ 1 


+ 12 
+ 24 


- 5 


- 9 


- 1 


+ 2 


- 2 




- 17 


• • • 


- 3 


- 4 
+ 1 


+ 1 




1 

1 

1 

1 

1 

1 

1 


•f 2 
- 8 


- '2 

+ 1 


+ 2 

+ 2 


• • • 






- 3 








-32 


+ 11 


+ 8 


+ 1 




. 


i-16 


+ 9 


- 4 






+ 8 


- 1 


- 2 






+ 20 


- 6 


+ 1 












1 


- 8 
















1 


+ 1 

























^2 


1'3« 
6W 


12«3 
hi?d 


1»23 


1»3 
h'd 

+ 8 

- 1 

-2 

• • • 

• • • 

+ 1 


2* 
c* 

+ 2 

- 2 

+ 2 

-2 

+ 1 


P2» 


P2» 


1«2 

-8 
+ 1 




(«) 


+ 12 


+ 24 


32 


16 


+ 20 


+ 1 


(71) 


- 5 


-17 


+ 11 


+ 9 


- 6 




(62) 


- 9 


• • • 


+ 8 


4 


+ 1 




(53) 


+ 3 


+ 6 


- 3 


+ 1 




(61) 
(521) 


+ 2 
+ 5 


4 


• • • 

- 5 






1 


+ 5 






1 


- 1 


- 3 


+ 1 








(431),- 2 


+ 1 


_ — 






(42-') ' + 1 




















i 














• 


1 













1 



































! 
































1 1 
! 






I 



















54—2 



428 



A KEHOIR ON THE SYMMETRIC FUNCTIONS 



[147 



IX (a). Runs on to p. 430. 



II 


9 

• 

- 1 


18 
6i 


27 

ch 


1'7 

- 1 

- 2 

- 15 
-72 


35 
dg 

- 1 

• • • 

- 5 

- 21 

-84 


125 
hcg 


1»6 
6V 


45 


135 
hdf 


2«5 


1»25 


1*5 


14» 


<9) 


















(81) 


















(72) 


















(63) 


















(64) 


















(71') 
















(621) 


















(531) 


















(4n) 


1 
I 


















(52') 


















(432) 


















(3') 


* 

- 1 
-9 


- 1 

- 7 
-36 


















(61') 


















(521') 


















(431«) 


















(42'1) 


















(3'21) 


















(32') 
















- 1 


(51*) 














1 


• • • 


(421») 












1 


4 


• • ■ 


(3n«) 










- 1 


2 


6 


• • • 


(32»1») 








- 1 


- 2 


5 


- 12 


- 2 


(2*1) 






- 1 


- 4 


- 6 


- 12 


- 24 


- 9 


(41-) 




- 1 


• • • 


• • • 


• • • 


5 


- 21 


• • ■ 


(321«) 


- 1 


- 3 


• • • 


- 4 


- 9 


- 23 


- 58 


- 6 


(2'1') 


- 3 


- 6 


- 3 


- 15 


- 24 


- 51 


- 108 


- 24 

1 


(31«) 


- 6 


- 19 


• • • 


- 15 


- 30 


- 81 


- 204 


- 20 


(2»1») 


- 17 


- 36 


- 10 


- 50 


- 81 


- 172 


- 366 


- 70 


(21') 


- 70 


-147 


- 35 


-161 


-252 


- 525 


- 1092 


-210 


(!•) 


-252 


-504 


- 126 


-504 


-756 


-1512 


-3024 


-630 



147] 



OF THE ROOTS OF AN EQUATION. 



429 



II 
(81 

(72; 

(63) 
(54; 

(71') 
(62i; 

(53i; 

(4n; 

(52«) 

(432) 

(3») 

(61») 

(521«) 

(431«) 

(42«i; 

(3«2i; 
(32' 
(51*) 

(42P) 
(3*1»J 

(32'1'^ 
(2*1 
(4r 

(321*) 
(2»1») 
(31') 

(2»1») 
(210 

(!•) 



234 
cde 


1»34 
b^de 


12H 

be'e 


1»24 
b^ce 


P4 
b^e 


3» 


123» 
6ccP 


P3» 

b^d" 


2»3 
c'd 


1«2«3 


































• 






























































































































1 


















1 


2 
















1 


• • • 


2 














1 


3 


3 


8 












1 


3 


6 


6 


15 










1 


• • • 


• • • 


• • • 


• • « 


• • • 








1 


5 


• • • 


• • • 


2 


• • • 


5 






1 


3 


10 


• • • 


2 


6 


7 


19 




1 


2 


7 


20 


• • • 


5 


17 


- 12 


36 


1 


2 


5 


- 12 


30 


3. 


- 13 


30 


- 27 


65 


3 


7 


- 12 


- 27 


60 


6 


- 27 


64 


- 51 


- 120 


■ • • 


• • ■ 


• • • 


4 


21 


• • • 


• • • 


6 


• • • 


12 


• • • 


3 


7 


- 25 


75 


• • • 


- 12 


42 


- 27 


85 


3 


6 


- 17 


- 42 


- 110 


6 


- 30 


72 


- 64 


- 152 


8 


- 19 


- 36 


85 


- 200 


- 15 


- 65 


- 152 


- 120 


- 281 


- 22 


- 48 


- 78 


- 168 


- 360 


- 36 


136 


- 300 


- 234 


- 516 


• • • 


- 10 


- 20 


- 75 


- 225 


• • ■ 


- 30 


- 110 


- 60 


- 200 


- 22 


54 


- 101 


- 241 


- 570 


- 36 


- 158 


- 372 


- 282 


- 656 


- 60 


- 129 


- 213 


- 459 


- 990 


- 93 


- 333 


- 720 


- 555 


- 1203 


- 60 


- 155 


- 270 


- 645 


- 1500 


- 90 


- 390 


- 920 

- 1740 


- 660 


- 1530 


- 166 


- 350 


- 565 


- 1200 


- 2550 


- 240 


- 820 


-1320 


- 2800 


- 455 


- 945 


- 1470 
-3780 


-3045 


- 6300 


- 630 


-2030 


- 4200 


-3150 


- 6510 


-1260 


-2520 


-7560 


15120 


- 1680 


-5040 


- 10080 


-7560 


- 15120 



4S0 



A MfcMOIR ON THE SYMMKTRIC FUNCTIONS 



[147 



II 


1*23 
h'cd 


P3 1 


12* 
6c* 


1*2» 
6V 


1»2« 
6V 


r2 


6» 


(9) 














1 


(81) 












1 


9 


(72) 










1 


7 


36 


(63) 








1 


5 


21 


84 


(54) 






1 


3 


10 


35 


126 


(7P) 




1 


• • • 


• • • 


2 


15 


- 72 


(621) 


1 


6 


• • • 


3 


17 


70 


252 


(531) 


4 
6 


15 


4 


15 


50 


161 


504 


(4n) 


20 


9 
6 


2i 


70 


210 


- 630 


(52«) 


9 


30 


24 


'- 81 


252 


756 


(432) 


22 


60 
90 


22 


60 


!- 165 


455 


- 1260 


(3«) 


- 36 


36 


93 


- 240 


630 


- 1680 


(6P) 


3 


19 


• • • 


6 


36 


147 


504 


(52 P) 


23 


81 


12 


51 


- 172 


525 


- 1512 


(43P) 


54 


- 155 


48 


- 129 


- 350 


945 


- 2620 


(42«1) 


- 101 


- 270 


78 


- 213 


i- 565 


- 1470 


- 3780 


(3«21) 


- 158 


390 


- 136 


- 333 


i- 820 


2030 


- 5040 


(323) 


- 282 


- 660 


- 234 


- 555 


- 1320 


- 3150 


- 7560 


(51*) 


58 


- 204 

- 645 


24 


- 108 


1- 366 


- 1092 


- 3024 


(42P) 


- 241 


- 168 


459 


- 1200 


- 3045 


- 7560 


(3»1») 


- 372 


- 920 


- 300 


- 720 


- 1740 


- 4200 


- 10080 


(32n«) 


- e.'ie 


1530 


516 


- 1203 


'- 2800 


- 6510 


- 15120 


(2*1) 


1140 


- 2520 


- 906 


- 2016 


1- 4500 


- 10080 


- 22680 


(4P) 


- 570 


- 1500 


- 360 


990 


- 2550 


- 6300 


- 15120 


(321*) 


1516 
- 2610 


- 3480 


- 1140 


- 2610 


- 5940 


- 13440 


- 30240 


(2»P) 


- 5670 


- 2016 


- 4383 


- 9540 


- 20790 


- 45360 


(3P) 


- 3480 


- 7800 


- 2520 


- 5670 


1- 12600 


- 27720 


- 60480 


(2»P) 


- 5940 


- 12600 


- 4500 


- 9540 


- 20220 

- 42840 


- 42840 


- 90720 


(2r) 


- 13440 


- 27720 


- 10080 


- 20790 


- 88200 


- 181440 


(!•) 


- 30240 


- 60480 


- 22680 


- 45360 


- 90720 


- 181440 


- 362880 



1 47] 



OP THE ROOTS OF AN EQUATION. 



431 



IX ijb). Runs on to p. 432. 



— 


9 

• 

J 


18 
bi 


27 
ch 


V7 


35 

d(/ 


125 
beg 


1'6 
b'g 
+ 9 

- 1 
-2 
-3 
-9 
+ 1 
+ 3 
+ 4 
+ 5 
+ 2 
+ 5 

• • • 

- 1 
-4 
-5 
-5 

• • • 

• • • 

+ 1 
+ 5 

• • • 

• « • 

• • • 

- 1 


45 


135 
bdf 


2«5 
-9 


1«25 
b'cf 


1*5 

hV 
-9 

+ 1 

+ 2 

+ 3 

+ 4 

- 1 
-3 
-4 

• * • 
-2 

• • • 

• • • 

+ 1 
+ 4 

• • • 

• • • 

• • • 

• • • 

- 1 


14« 
b^ 


234 

cde 


1*34 


(9) 


- 9 


+ 9 


+ 9 


- 9 


+ 9 


- 18 


+ 9 
- 9 


- 18 


+ 27 


- 9 


- 18 


+ 27' 


(81) 


+ 9 


- 1 


- 9 


+ 1 


- 9 


+ 10 


+ 10 


+ 9 


- 11 


+ 5 


+ 18 
+ 4 


- 11 


(72) 


+ 91- 9 


+ 5 


+ 2-9 


+ 4 


- 9 


+ 18 


-5 
+ 9 


- 6 


+ 9 


-20 


(63) 


+ 9 


- 9 


- 9 


+ 9 


+ 9 


• • • 


-. 9 


• • • 


- 9 


+ 9 


• • • 


- 9' 


(54) 


+ 9 
- 9 


- 9 


- 9 


+ 9 


- 9 


+ 18 


+ 11 


- 2 


- 1 


- 7 


-11 


- 2 


+ 13 
+ 11 


(71') 


+ 1 


+ 2 


- 1 


+ 9 


- 3 


+ 9 


- 10 


-2 


+ 4 


- 5 


- 11 


(621) 


-18 1+10 


+ 4 


- 3 


• • • 


- 8 


+ 18 


- 10 


- 4 

-8 


+ 11 


- 14 


- 4 

+ 2 


+ 13i 


(531) 


- 18'+ 10 


+ 18 


- 10 


• • • 

+ 9 


- 10 


- 2 


- 5 


+ 15 


+ 6 


- Tj 


(4«1) 


- 9 


+ 5 


+ 9 


- 5 


- 14 


- 11 


+ 6 


+ 1 

• • • 

+ 6 
-3 
+ 2 
+ 6 
• • • 
-6 
+ 3 

• • • 

-2 

+ 2 


- 1 


+ 5 


+ 2 


- 5 


(52') 


-9+9 


- 5 


- 2 


+ 9 


- 4 


- 1 


- 8 
+ 2 


+ 6 
- 6 


+ 1 


+ 6 


• « • 

+ 1 


(432) 


-181+18 


+ 4 


- 11 


• • • 


- 4 


- 2 


+ 2 


- 8 
+ 3 
+ 5 


(3'; 


- 3 


+ 3 
- 1 


+ 3 
- 2 


- 3 


- 6 


+ 3 


+ 3 


+ 3 


• • • 


- 3 


• • • 


(61') 


+ 9 


+ 1 


- 3 

- 9 


+ 3 


- 9 


+ 4 


- 4 


+ 5 


- 5 


(i521») 


+ 27 


- 11 


- 6 


+ 4 
+ 11 
+ 5 


+ 11 


- 7 
+ 13 
+ 3 


+ 15 


- 15 


- 1 


- 5 


+ 1 


(m') 


+ 27 


- 11 


- 20 
+ 1 

- 13 


- 9 

- y 


+ 13 
+ 12 


- 5 

- 2 

- 4 
+ 2 


+ 1 
+ 3 


- 5 


+ 1 


+ 2 


(+-«1) 


+ 27 


- 19 


+ 1 


+ 2 


- 11 

1 


(:V''21) 


+ 27' 


- 19 

- 9 


+ 12 
+ 2 


+ 18 


. — 7 

1 


- 7 


• • • 


+ 3 


- 1 


(32») 


+ 9 


+ 5 


- 3 
+ 3 


- 2 


+ 1 


• • a 


- 1 




(51') 


- 9 


+ 1 


+ 2 


- 1 

- 5 


- 3 


+ 4 


- 4 


+ 4 


1 


(421») 


-36 
- 18 


+ 12 


+ 8 


+ 12 


- 14 


- 4 


+ 1 


- 1 






(3n») 


+ 6 


+ 11 


- 6 


- 3 


+ 1 
+ 4 


- 2 


+ 2 


- 1 










(32n») 


-54 


+ 30 


+ 5 
— 5 


- 9 


- 9 
+ 3 


+ 4 
- 1 


- 1 




1 






(2'1) 


- 9 


+ 7 


• • • 


• • • 






1 


. 




(41») 


+ 9 
+ 45 


- 1 


- 2 


+ 1 


- 3 


+ 3 










1 




(321*) 


- 13 


- 10 


+ 6 


+ 3 


- 1 















(2'1») 


+ 30 


- 14 


+ 5 


• • ■ 


- 1 
















1 


(31*) 


- 9!+ 1 


+ 2 
- 1 


- 1 






" 














(2'1») 


- 27 


+ 7 














1 


(210 


+ 9 


- 1 






! 








1 1 


(!•) 


- 1 
















1 



432 



A MEMOIR ON THE SYMMETRIC FUNCTIONS 



[147 



__ 


12«4 
bi^e 


P24 
b^ce 


P4 
b'^e 

+ 9 

- 1 
-2 
-3 

• • • 

+ 1 
+ 3 

- 1 


3» 
(P 

-3 

+ 3 

+ 3 

-6 

+ 3 

-3 

+ 3 

+ 3 

-3 

-3 

+ 3 

- 1 


123» 
bccP 


P3» 
b^(P 


2»3 

+ 9 
-9 
+ 5 
-3 
+ 1 
+ 2 
- 2 
+ 2 
-1 


1«2>3 


1*23 
b*cd 


1*3 
b*d 

-9 

+ I 
+ 2 

• • • 
« • • 

- 1 


12* 
be* 

-9 

+ 7 

-5 

+ 3 

- 1 


1«2» 
6V 


1»2» 


V2 
b'c 

+ 9 

-1 


I* 

-1 


(9) 


+ 27 


-36 


+ 27 


- 18 


-54 


+ 45 


+ 30 


-27 


(81) 


- 19 


+ 12 


-19 


+ 6 


+ 30 


-13 


-14 


+ 7 


(72) 


+ 1 


+ 8 


- 13 


+ 11 


+ 5 


-10 


+ 5 


- 1 


(63) 


- 9 


+ 12 


+ 18 


- 3 


- 9 


+ 3 


- 1 




(54) 


+ 3 


- 4 


- 7 


- 2 


+ 4 


• • • 






(7P) 


+ 5 


- 5 


+ 12 


- 6 


- 9 


+ 6 






(621) 


+ 12 


- 14 
+ 1 


- 7 


+ 1 


+ 4 


- 1 










(531) 


- 2 


- 4 


+ 2 


- 1 








(in) 


+ 1 


• • • 


+ 3 


• • • 










(52«) 


- 6 


+ 2 


+ 3 


- 1 










(432) 


+ 2 


• • a 


- 1 
















(3») 


• • • 


• • • 


















(61') 


- 5 


+ 5 














(52P) 


+ 3 


- 1 














(43 1«) 


- 1 


















































































































































































































i 

1 














































• 










! 
















1 













147] 



OF THE ROOTS OF AN EQUATION. 



433 



X (a). Runs on to p. 436. 



II 


10 
k 


19 


28 
ci 


P8 
hH 


37 
dh 


127 
hch 


1»7 
6»A 


46 
^9 


136 
hdg 


2»6 


P26 
h'cg 


P6 
h'g 


5» 


(10) 






















(91) 
























(82) 






















(73) 






















(64) 
























(5') 




















(8P) 






















(721) 


+ 1 




















(631) 




















(541) 




















(62') 






















(532) 




















(4'2) 






















(43») 




















(71') 




















(62 1») 




















(53P) 


















• 


(4n') 






















(52=1) 




















(4321) 
























(3'1) 




















(42') 




















(3»2') 




















(6P) 




















(52P) 




















(431') 


















, 




(42n») 




















(3»2P) 




















(32'1) 




















(2») 


















+ 1 


(5P) 
















+ 1 


• • • 


(421«) 






+ 1 

+ 2 
+ 17 
+ 90 














+ 1 


+ 4 


• • • 


(3'P) 














-f 1 


+ 2 


+ 6 


• • • 


(32'1») 










+ 1 


+ 2 


+ 5 


+ 12 


• • • 


(2*1') 








+ 1 


+ 4 


+ 6 


+ 12 


+ 24 


+ 2 


(41») 






+ 1 


• • • 


• • • 


■ • • 


-f 6 


+ 25 


• • • 


(321») 




+ 1 


+ 3 


• • • 


-f 5 


+ 11 


+ 28 


+ 70 


• • • 


(2«1') 


+ 1 


+ 3 


+ 6 


+ 4 


+ 19 


+ 30 


+ 63 


+ 132 


+ 6 


(310 


• • • 


+ 7 


+ 22 


• • • 


+ 21 


-f 42 


-f 112 


+ 280 


• • • 


(2'1') 


+ 1 

-f 8 
+ 45 


+ 6 


+ 20 


+ 42 


+ 15 


+ 72 


+ 115 


+ 242 


+ 510 


+ 20 


(21») 


+ 1 
+ 10 


+ 28 


+ 92 


+ 192 


-f 56 


+ 252 


+ 392 


+ 812 


+ 1680 


+ 70 


(1") 


+ 120 


-f 360 


-f 720 


+ 210 


+ 840 


+ 1260 


+ 2520 


+ 5040 


+ 252 : 



C. II. 



55 



434 



A MEMOIB OK THE STMMETBIC FUNCTIONS 



[147 



II 


145 


235 
cdf 


1«35 


12»5 
hi?f 


1»25 
hhf 


P5 


24« 


1»4» 
6V 


3>4 

d^e 


1234 
hcde 


(10) 






















(91) 






















(82) 






















(73) 






















(64) 






















(5') 






















(81') 






















(721) 






















(631) 






















(541) 






















(62') 






















(632) 






















(4«2) 






















(43«) 






















(71') 






















(621') 






















(531') 






















(4«1') 






















(52*1) 






















(4321) 




















+ 1 


(3'1) 


















+ 1 


+ 3 


(42») 
















+ 1 


« • • 


+ 3 


(3»2») 














+ 1 


+ 2 


+ 2 


+ 8 


(61') 












+ 1 


• • • 


• • • 


• • • 


• • • 


(521») 










+ 1 


+ 5 


• • • 


• • • 


• • « 


m • • 


(431') 








+ 1 


+ 3 


-f 10 


• • • 


• • • 


• • • 


+ 3 


(42'1») 






+ 1 


+ 2 


+ 7 


+ 20 


• • • 


+ 2 


• • • 


+ 8 


(3'21') 




-f 1 


+ 2 


+ 5 


+ 12 


+ 30 


+ 2 


+ 4 


+ 5 


+ 21 


(32*1) 


+ 1 


+ 3 


+ 7 


+ 12 


+ 27 


+ 60 


+ 7 


+ 16 


+ 12 


+ 49 


(2») 


+ 5 


+ 10 


+ 20 


+ 30 


+ 60 


+ 120 


+ 20 


+ 45 


+ 30 


+ 110 


(51") 


• • • 


• • • 


• * • 


... 


+ 5 


+ 26 


• • • 


• • • 


• • • 


• • • 


(421') 


• • • 


• • • 


-f 4 


+ 9 


+ 32 


+ 95 


• • • 


+ 6 


• • • 


+ 22 


(3»1*) 


• • • 


+ 4 


-f 8 


+ 22 


+ 54 


-f 140 


+ 6 


+ 12 


+ 12 


+ 56 


(32»1') 


-f 3 


+ 11 


-f 26 


-f 48 


+ 112 


-f 260 
+ 480 


+ 18 


+ 42 


+ 31 


+ 128 


(2n') 


+ 14 


-f 32 


+ 68 


+ 108 


+ 228 


+ 53 


+ 114 


+ 80 


+ 284 


(41«) 


• • • 


• • • 


+ 15 


+ 30 


+ 111 


+ 330 


• • • 


+ 20 


• • • 


+ 60 


(321») 


+ 10 


+ 35 


4- 85 


+ 156 


+ 368 


+ 860 


+ 50 


+ 120 


+ 80 


+ 335 


(2'1<) 


+ 42 


+ 99 


+ 210 


+ 339 


+ 720 


+ 1530 


+ 144 


+ 306 


+ 213 


+ 735 


(310 


+ 35 


+ 105 


-f 266 


+ 462 


+ 1092 


+ 2620 


+ 140 


+ 360 


+ 210 


+ 876 


(2»1«) 


+ 130 


+ 296 


+ 622 


+ 990 


+ 2082 


+ 4380 


+ 400 


+ 840 


+ 570 


+ 1900 


(21') 


+ 406 


+ 868 


-f 1792 


+ 2772 


+ 5712 


+ 11760 


+ 1120 


+ 2310 


+ 1540 


+ 4900 


(1") 


+ 1260 


+ 2520 


+ 5040 


+ 7560 


+ 15120 


+ 30240 


+ 3150 


+ 6300 


+ 4200 


+ 12600 



147] 



OF THE ROOTS OF AN EQUATION, 



435 



P34 


2»4 


1«254 
6Vc 


1*24 


1«4 


13' 


223» 


1>23« 


1*3» 
h'd^ 


12»3 
h&d 


















































































































































































• 






















-f 1 


















+ 1 


• • • 
















+ 1 


+ 4 


+ 3 














+ 1 


+ 2 


+ 6 


+ 7 












+ 1 


-f 2 


+ 5 


+ 12 


+ 12 










+ 1 




• • • 


• • • 


• • • 


• • • 








+ 1 


+ 6 




• • • 


• • • 


+ 2 


• • • 






+ 1 


+ 4 


+ 15 




• • • 


+ 2 


+ 8 


+ 7 




+ 1 


-f 2 


-f 6 


+ 20 




+ 2 


+ 4 


+ 12 


+ 16 


+ 1 


■ • • 


-f 2 


+ 9 


+ 30 




• • • 


+ 5 


+ 22 


-f 12 


+ 3 


+ 3 


+ 8 


+ 22 


+ 60 


+ 3 


+ 8 


+ 21 


+ 56 


+ 49 

-f 87 


+ 6 


+ 6 


+ 15 


+ 36 


+ 90 


-f 10 


+ 18 


+ 42 


+ 96 


+ 10 


+ 6 


+ 18 


+ 48 


+ 120 


-f 6 


+ 15 


+ 42 


+ 115 


+ 87 


+ 18 


+ 15 


+ 34 


+ 78 


+ 180 


+ 18 


+ 34 


+ 80 


+ 188 


+ 156 


• • • 


• • • 


• • • 


+ 4 


-f 25 


• • • 


• • • 


• • • 


+ 6 


• • • 


+ 3 


• • • 


+ 7 


+ 32 

+ 76 


+ 111 


• • • 


• • • 


+ 12 


+ 54 


+ 27 


+ 9 


+ 10 


+ 27 


+ 215 


+ 6 


-f 18 


+ 48 


+ 132 


+ 112 


+ 27 


+ 18 


+ 54 


+ 149 


-f 390 


+ 15 


+ 34 


-f 99 


+ 270 


-f 198 


+ 48 


+ 42 


+ 99 


+ 236 


+ 570 


-f 42 


+ 80 


+ 186 


+ 436 


+ 358 


+ 112 


+ 87 


+ 198 


+ 450 


+ 1020 


+ 87 


-f 156 


+ 358 


+ 820 


+ 645 


+ 240 


+ 180 


+ 390 


+ 840 


+ 1800 


+ 180 


+ 310 


+ 680 


+ 1500 


+ 1170 


+ 10 


• • • 


+ 20 


+ 95 


-f 330 


• • • 


• • • 


-f 30 


+ 140 


+ 60 


+ 76 


+ 48 


-f 149 


+ 416 


+ 1095 


+ 36 


+ 78 


+ 236 


-f 650 


+ 450 


+ 132 


+ 115 


+ 270 


+ 650 


+ 1580 


+ 96 


-f 188 


+ 436 


+ 1032 


+ 820 


+ 294 


+ 228 


+ 523 


+ 1196 


+ 2730 


+ 210 


+ 370 


+ 844 


+ 1920 


+ 1479 


+ 612 


+ 468 


+ 1008 


+ 2172 


+ 4680 


+ 444 


+ 740 


+ 1604 


+ 3480 


+ 2688 


+ 215 


+ 120 


+ 390 


+ 1095 


+ 2850 


+ 90 


+ 180 


+ 570 


+ 1580 


+ 1020 


+ 775 


+ 585 


+ 1340 


+ 3050 


+ 6900 


+ 510 


+ 880 


+ 2000 


+ 4520 


+ 3390 


+ 1566 


+ 1194 


+ 2547 


+ 5436 


+ 11610 


+ 1092 


+ 1776 


-h 3792 


+ 8100 


+ 6180 


+ 2030 


+ 1470 


+ 3360 


+ 7560 


-f 16800 


+ 1260 


+ 2100 


+ 4760 


+ 10640 


+ 7770 


+ 3990 


+ 3015 


+ 6330 


-f 13290 


+ 27900 


-f 2700 


+ 4280 


+ 8980 


-f 18840 


+ 14220 


+ 10080 


+ 7560 


+ 15540 


+ 31920 


+ 65520 


+ 6720 


+ 10360 


+ 21280 


+ 43680 


+ 32760 


+ 25200 


+ 18900 


+ 37800 


+ 75600 + 151200 


4- 16800 


-f 25200 


-f 50400 


-f 100800 


+ 75600 



55-2 



436 



A MEMOIR ON THE SYMMETRIC FUNCTIONS 



[147 



II 


i 


l»2«3 
9c'd 




P23 
b^cd 




V3 




2» 




2»3> 
b^c* 




P2» 
6V 




1«2» 
6V 




1»2 




110 

6" 




(10) 


















4- 


1 




(91) 
















4- 


1 


4- 


10 




(82) 














4- 


1 


4- 


8 


4- 


45 




(73) 




• 








4- 


1 


4- 


6 


4- 


28 


4- 


120 




(64) 










4- 


1 


4- 


4 


4- 


15 


4- 


56 


4- 


210 




(5') 








4- 


1 


4- 


2 


4- 


6 


4- 


20 


+ 


70 


4- 


252 




(81') 






4- 
4- 


1 

t 


• • • 


• • • 


• • • 


4- 


2 


4- 


17 


4- 


90 




(721) 




+ 


1 


• • • 


• • • 


4- 


3 


4- 


20 


4- 


92 


4- 


360 




(631) 


+ 


1 


+ 


5 


4- 


21 


• • • 


4- 


4 


4- 


19 


4- 


72 


+ 


252 


4- 


840 




(541) 


+ 


3 


+ 


10 


4- 


35 


4- 


5 


4- 


14 


4- 


42 


4- 


130 


4- 


406 


4- 


1260 




(62») 


+ 


2 


■+ 


11 


4- 


42 


• • • 


4- 


6 


4- 


30 


4- 


115 


4- 


392 


4- 


1260 




(532) 


+ 


11 


+ 


35 


4- 


105 


4- 


10 


4- 


32 


4- 


99 


4- 


296 


4- 


868 


4- 


2520 




(4*2) 


+ 


18 


+ 


50 


4- 


140 


4- 


20 


4- 


53 


4- 


144 


4- 


400 


4- 


1120 


4- 


3150 




(43') 


+ 


31 


+ 


80 


4- 


210 


4- 


30 


4- 


80 


4- 


213 


4- 


570 


4- 


1540 


+ 


4200 




(7P) 


• • • 


+ 


3 


4- 

4- 


22 
112 


• • • 


• • • 


4- 


6 


4- 


42 


4- 


192 


4- 


720 




(62 1») 


+ 


5 


+ 


28 


• • • 


4- 


12 


4- 


63 


4- 


242 


4- 


812 


4- 


2520 




(sai*) 


+ 


26 


+ 


85 


4- 


266 


4- 


20 


4- 


68 


4- 


210 


4- 


622 


+ 


1792 


4- 


5040 




(4'1») 


+ 


42 


+ 


120 


4- 


350 


4- 


45 


4- 


114 


4- 


306 


4- 


840 


4- 


2310 


4- 


6300 




(52n) 


+ 


48 


+ 


156 


4- 


462 


4- 


30 


4- 


108 


4- 


339 


4- 


990 


4-' 


2772 


4- 


7560 




(4321) 


+ 


128 


+ 


335 


4- 


875 


4- 


110 


4- 


284 


4- 


735 


4- 


1900 


4- 


4900 


4- 


12600 




(3'1) 


+ 


210 


+ 


510 


4- 


1260 


4- 


180 


+ 


444 


4- 


1092 


4- 


2700 


4- 


6720 


4- 


16800 




(42') 


+ 


228 


+ 


585 


4- 


1470 


4- 


180 


4- 


468 


4- 


1194 


4- 


3015 


4- 


7560 


4- 


18900 




(3»2'') 


+ 


370 


+ 


880 


4- 


2100 


4- 


310 


4- 


740 


4- 


1776 


4- 


4280 


4- 


10360 


4- 


25200 




(61') 


+ 


12 


+ 


70 


4- 


280 


• • • 


4- 


24 


4- 


132 


4- 


510 


4- 


1680 


4- 


5040 




(521») 


+ 


112 


4- 


368 


4- 


1092 


4- 


60 


4- 


228 


4- 


720 


4- 


2082 


4- 


5712 


4- 


15120 




(43P) 


+ 


294 


+ 


775 


4- 


2030 


4- 


240 


4- 


612 


4- 


1566 


4- 


3990 


4- 


10080 


4- 


25200 




(42n») 


+ 


523 


+ 


1340 


4- 


3360 


4- 


390 


4- 
4- 


1008 


4- 


2547 


4- 


6330 


4- 


15540 


4- 


37800 




(3»21') 


+ 


844 


+ 


2000 


4- 


4760 


4- 


680 


1604 


4- 


3792 


4- 


8980 


4- 


21280 


4- 


50400 




(324) 


+ 


1479 


+ 


3390 


4- 


7770 


4- 


1170 


4- 


2688 


4- 


6180 


4- 


14220 


4- 


32760 


4- 


75600 




(2') 


+ 


2580 


+ 


5700 


4- 


12600 


4- 


2040 


4- 


4530 


4- 


10080 


4- 


22500 


4- 


50400 


4- 


113400 




(51') 


+ 


260 


4- 


860 


4- 


2520 


4- 


120 


4- 


480 


4- 


1530 


4- 


4380 


4- 


11760 


4- 


30240 




(121*) 


+ 


1196 


+ 


3050 


4- 


7560 


4- 


840 
1500 


+ 


2172 


4- 


5436 


4- 


13290 


4- 


31920 


4- 


75600 




(3»1*) 


4- 


1920 


4- 


4520 


4- 


10640 


4- 


4- 


3480 


4- 


8100 


4- 


18840 


4- 


43680 


4- 


100800 




(32n>) 


+ 


3358 


4- 

4- 


7610 
12720 


4- 
4- 


17220 4- 


2580 


; 4- 


5844 


4- 


13212 


4- 


29820 


4- 


67200 


4- 


151200 




(2n») 


+ 


5844 


27720 


4- 


4530 


4- 


9876 


4- 


21564 


4- 


47160 


4- 


103320 


4- 


226800 




(41«) 


+ 


2730 


4- 


6900 


16800 


4- 


1800 


4- 


4680 


+ 


11610 


4- 


27900 


4- 


65520 


4- 


151200 




(321*) 


+ 


7610 


4- 
4- 


17000 
28260 


4- 


37800 1 4- 


5700 


+ 


12720 


4- 


28260 


4- 


62520 


4- 


137760 


4- 


302400 




(2»1*) 


-f 


13212 


4- 


60480 4- 


10080 


4- 


21564 


+ 


46152 


4- 


98820 


4- 


211680 


4- 


453600 




(31') 


+ 


17220 


4- 


37800 


4- 


82320 


4- 


12600 


4- 


27720 


4- 


60480 


4- 


131040 


4- 


282240 


4- 


604800 




(2n») 


+ 


29820 


4- 


62520 


4- 
4- 
4- 


131040 


+ 


22500 


4- 


47160 


4- 


98820 


4- 


207000 


+ 


433440 


4- 


907200 




(21') 


67200 


4- 


137760 
302400 


282240 


4- 


50400 


4- 


103320 


4- 


211680 


4- 


433440 


4- 


887040 


4- 


1814400 




(V) 


+ 


151200 


4- 


604800 


4- 


113400 


4- 


226800 


4- 


453600 


4- 


907200 


4- 


1814400 1 + 


362880C 


ii 



147] 



OF THE BOOTS OF AN EQUATION. 



437 





10 

k 


19 


28 
ci 


P8 

bH 


£(6). 

37 

dh 


Run 

127 
bch 

-20 


s on 1 
P7 


ko p. - 
46 


139. 

136 
bdg 


2>6 


P26 
b^'cg 

4- 30 

- 12 


1*6 
b'g 


5» 


145 
be/ 


(10) 


10 


+ 10 


+ 10 


- 10 


+ 10 


4- 10 


4- 10 


-20 


- 10 


-10 


+ 


5 


20 


(91) 


+ 10 


- 1 


10 

4- 6 


4. 1 


- 10 


4- 11 


- 1 

- 2 


- 10 


4- 11 


4- 10 


4- 1 

4. 2 


— 


5 


+ 11 


(82) 


+ 10 


- 10 


4- 2 


- 10 


4- 4 


10 


4- 20 


- 6 


- 6 


— 


5 


+ 20 


(73) 


+ 10 


- 10 


- 10 


4- 10 


4- 11 


- 1 


- 3 


- 10 


- 1 


4- 10 


- 9 


4- 3 


— 


5 


+ 20 

- 4 


(64) 


+ 10 


- 10 

- 5 


- 10 


4- 10 


- 10 


4- 20 


- 10 


4- 14 


- 4 


- 2 


- 6 


4- 4 


4- 


5 
10 


(5*) 


+ 5 


- 5 


4- 5 


- 5 


4- 10 


5 


5 


4- 10 


4- 5 


- 15 


+ 5 


- 15 


(8P) 


10 


+ 1 


^ ^ 


- 1 


4- 10 


- 3 


4- 1 


4- 10 


- 11 


- 2 

- 4 


+ 4 


- 1 


4- 


5 


- 11 
-31 

7 


(721) 


- 20 


+ 11 


4- 4 


- 3 


- 1 


- 8 


4- 3 


+ 20 


- 10 


4- 11 


- 3 

- 4 


4- 


10 


(631) 


- 20 


+ 11 
+ 11 


4- 20 
+ 20 


- 11 


- 1 


-10 


+ 4 


4 


- 4 


- 8 


4- 15 


4- 


10 
15 


(541) 


- 20 


- 11 

- 2 


4- 20 


-31 


+ 11 


- 4 


- 7 


- 8 


4- 18 
4- 6 


- 5 


4- 23 


(62*) 


- 10 


+ 10 


- 6 


+ 10 


- 4 


4- 2 


- 2 


- 8 


• • ■ 


- 2 

- 5 


+ 


5 


- 8 


(532) 


- 20 


4- 20 


4- 4 
4- 2 


- 12 


1 


- 3 


4- 5 


4- 20 


- 19 


- 4 

4. 10 


4. 15 


— 


15 


+ 10 


(4«2) 


- 10 


+ 10 


- 6 


4- 10 


- 12 


+ 6 

4- 3 


14 


4- 4 


- 6 


• • • 


+ 


5 


4- 4 


(43«) 


10 


+ 10 
- 1 


+ 10 


- 10 


- 11 


4- 1 


- 2 


4. 13 


- 4 


- 3 


• • • 


4- 


5 


- 8 


(7P) 


+ 10 


2 
-6 


4- 1 

4- 4 


- 3 


4. 3 


- 1 


- 10 


4. 4 


4- 2 


- 4 


4. 1 


— 


5 


4. 11 


(62P) 


+ 30 


- 12 


9 


+ 11 


- 4 


- 6 


4- 15 


4. 6 


15 


4- 4 


— 


15 


4- 18 


(53P) 


+ 30 


- 12 


-22 
- 11 

4- 2 


4. 12 


- 9 


4. 13 


5 


- 6 


4- 15 


4- 10 


- 19 


4- 5 


4- 


10 


- 12 


(4«P) 


+ 15 


- 6 


4- 6 


- 15 


4- 17 


- 6 


4- 9 


- 3 


- 1 


+ 1 


• • • 


4- 


5 


- 8 


(52«1) 


+ 30 


-21 


4- 5 


- 9 


4- 12 


- 5 


- 18 


4- 18 


+ 4 


-17 


4- 5 


4- 


10 


- 1 


(4321) 


+ 60 


-42 


-28 


4- 26 


4. 3 


4- 21 


- 12 


4- 12 


- 15 


- 8 


4- 7 


• • • 


— 


5 


• • • 


(3'1) 


+ 10 


7 


- 10 


4- 7 


+ 11 


- 4 


• • • 


4- 2 


- 7 

• • • 


4- 4 


• • • 


• • • 


- 5 


4- 5 


(42») 


+ 10 


- 10 


4- 6 


4- 2 


- 10 


4- 4 


- 2 


+ 10 


- 4 


4- 2 


• • • 


— 


5 


• • • 


(3«22) 


4- 15 


- 15 
+ 1 
+ 13 
+ 13 


4- 1 


4- 7 


4- 6 


- 7 


• • • 


- 9 


4- 3 


4- 2 


• • • 


• • • 


4- 


5 


- 1 


(6P) 


- 10 


4- 2 

+ 8 


- 1 


+ 3 


- 3 


+ 1 


4- 4 


- 4 


- 2 


4- 4 


- 1 


4- 


5 


- 5 


(521») 


- 40 


- 5 


+ 12 


- 14 

- 16 

- 23 


+ 5 


4. 16 


-19 


- 8 


4. 19 


- 5 


— 


5 


4- 1 


(43P) 


- 40 


+ 24 


- 13 

- 9 
-21 


4- 12 


+ 6 


- 8 


4- 5 


• • • 


- 1 




— 


5 


4- 5 


(42212) 


- 60 


4- 33 
+ 33 


4- 4 


4- 18 


+ 9 


- 12 


4- 3 


+ 8 


4 




4- 


5 


- 1 


(322P) 


- 60 


4- 28 


-24 


4. 9 


• • • 


• • • 


4- 6 


- 4 


• • • 




4- 


5 


- 3 


(32n) 


- 40 


+ 31 


- 8 

- 2 


- 7 


- 2 


4- 5 


• • • 


4. 8 


- 3 


• • • 


■ • • 




— 


5 


4- 1 


(2») 


- 2 


+ 2 


• • • 


4. 2 


• • • 


• • • 


- 2 


• • • 


• • • 


• • • 




4- 


1 




(5P) 


+ 10 


- 1 


2 


+ 1 


- 3 


+ 3 


- 1 


- 4 


4. 4 


4- 2 


- 4 


4- 1 






(42P) 


+ 50 


- 14 


- 10 


+ 6 


-15 


4- 17 


- 6 


4- 4 


1 


2 


4- 1 








(3«P) 


+ 25 


- 7 


- 13 


4. 7 


4- 3 

4- 12 


- 1 


• • • 


4- 2 


- 2 


4- 1 










(32n») 


+ 100 


46 


- 12 


4. 14 


- 5 


• • • 


- 4 


4. 1 












(2n') 


+ 25 


- 16 


+ 9 

4- 2 


• • ■ 


- 4 
+ 3 


• • • 


• • • 


4- 1 














(4P) 


- 10 


+ 1 


- 1 


- 3 


+ 1 















(32P) 


- 60 


4- 15 


+ 12 


- 7 


- 3 


4- 1 

















(2'P) 


- 50 


+ 20 


6 


• • • 


4. 1 


















(3F) 


+ 10 


1 


- 2 


+ 1 






















(2n«) 


+ 35 


- 8 


4- 1 
























(2P) 


- 10 


4. 1 










1 
1 














(PO) 


+ 1 























438 



A MEMOIR ON THE SYMMETRIC FUXCTIONS 



[147 



— 


235 
cdf 


1>35 
hHf 

+ 30 

12 


12«5 
6cy 


P25 


1»5 


24» 

C6* 


1H» 
6V 


3«4 


1234 
bcde 

+ 60 

-42 


P34 
b^de 


2»4 

c'e 


1»2H 


1*24 
b^ce 


b^e 


(10) 


-20 
+ 20 


+ 30 


-40 


+ 10 


- 10 


+ 15 


- 10 
+ 10 


-40 


+ 10 


-60 


+ 50 


-10 


(91) 


-21 


+ 13 


- 1 


+ 10 
+ 2 


- 6 


+ 13 


-10 


+ 33 


-14 


+ 1 


(82) 


+ 4 


-22 


+ 2 


+ 8 


- 2 


- 11 


+ 10 
-11 
- 2 

+ 5 


-28 


+ 24 
+ 12 


+ 6 


+ 4 


- 10 


+ 2 


(73) 


- 1 


- 9 


- 9 


+ 12 


- 3 


+ 10 


-15 


+ 3 
+ 12 


- 10 


+ 18 


- 15 


+ 3 


(64) 


+ 20 


- 6 


- 18 


+ 16 


4 


- 14 

+ 5 


+ 9 
+ 5 


- 8 


+ 10 
- 5 


- 12 


+ 4 


• • • 

• • • 

- 1 

- 3 


(5^) 


- 15 


+ 10 


+ 10 


- 5 


• • • 


- 5 


- 5 


+ 5 


• • • 


(8P) 


- 12 


+ 12 


+ 5 


- 5 


+ 1 
+ 3 


- 6 


+ 6 


- 10 


+ 26 


-13^ 


+ 2 


- 9 


+ 6 


(721) 


- 3 


+ 13 


+ 12 


- 14 


- 12 


+ 17 


+ 1 


+ 21 


- 16 


+ 4 


-23 


+ 17 


(631) 


- 19 


+ 15 


+ 18 


- 19 


+ 4 


+ 4 
+ 4 


- 3 


+ 13 


- 15 


+ 5 


• • • 

• • • 


+ 3 


- 1 


(541) 


+ 10 


- 12 


- 1 


+ 1 


• • • 


- 8 


8 


• • • 


+ 5 


- 1 


• ■ • 


(62») 


- 4 


+ 10 
- 4 
+ 2 


+ 4 


- 8 


+ 2 


+ 10 


- 1 


- 4 
+ 1 

; 2 


- 8 
+ 5 
+ 4 

- 3 


• • • 


- 4 


+ 8 


- 2 


(532) 


+ 17 


- 13 


+ 5 


• • • 


- 12 


+ 1 

3 

+ 3 

- 6 


- 1 

• • • 


+ 4 


- 2 


■ • • 


(4«2) 


- 12 


+ 2 


• • • 

• • • 

+ 5 
+ 19 


• • • 

• • • 


+ 2 
+ 2 
+ 6 
- 6 
+ 2 


- 2 


• • • 


• • • 


(43«) 


+ 1 


- 3 


+ 3 


- 1 


• • • 


• • • 


• • • 


• ■ • 


+ 1 


(7P) 


+ 5 


- 5 


- 5 

- 17 


- 1 

- 4 


+ 3 


12 
+ 7 


+ 6 


- 2 


+ 9 


- 6 


(621«) 


+ 15 
- 4 


- 19 


+ 1 


- 3 

- 3 


- 1 


+ 2 
- 2 


- 4 


+ 1 


(53 1«) 


+ 3 


+ 2 


1 


+ 2 


+ 4 


- 2 


+ 1 






(4n«) 


-f 1 


+ 2 


- 1 


• • • 




- 3 


+ 3 


+ 3 


- 3 


• • • 


+ 1 








(52n) 


- 13 


+ 2 


+ 9 
- 3 


- 3 

• • • 

• • • 

• • • 

• • • 

- 5 
+ 1 




+ 2 


- 1 

- 3 


+ 3 


- 3 


+ 1 








(4321) 


+ 5 


+ 4 


+ 4 


- 3 
+ 1 


+ 1 








(3'1) 


- 1 


• • • 


• • • 


- 2 

- 2 
+ 1 


• • • 












(42') 


+ 4 


- 2 


■ • • 




+ 1 












(3>2*) 


- 2 

- 5 
+ 5 

- 1 

- 2 
+ 1 


• • • 


• • • 
















(61*) 


+ 5 


+ 5 


+ 1 
















(52 P) 


1 


- 3 






















(43P) 


2 


+ 1 




















(42n') 


+ 1 


















(3»21>) 









































































































































































































































































































































































































147] 



OF THE ROOTS OF AN EQUATION. 



439 



13» 
hd^ 


2'3« 
c^d^ 


1«23« 
h^cd^ 


1*3« 
h*d^ 


12»3 


P2«3 


P23 
h^cd 


r3 

Vd 


2» 

c» 

-2 

+ 2 
-2 

+ 2 
-2 
+ 1 


2>3» 
+'25 


1*2» 
6V 


1«2» 
6V 


P2 


110 

+ 1 


+ 10 


+ 15 


-60 
+ 33 


+ 25 


-40 


+ 100 


-60 


+ 10 


-50 


+ 35 


- 10 


- 7 


- 15 


- 7 


+ 31 


- 46 


+ 15 


- 1 


- 16 


+ 20 


- 8 


+ 1 


- 10 


+ 1 


+ 28 


- 13 


- 8 


- 12 


+ 12 


- 2 


+ 9 


- 6 


+ 1 




-h 11 


+ 6 


- 24 


+ 3 


- 2 


+ 12 


- 3 


• • • 


- 4 


+ 1 






+ 2 


- 9 


• • • 


+ 2 


+ 8 


- 4 


• • • 


• • • 


+ 1 








- 5 


+ 5 


+ 5 


• • • 


- 5 


• ■ • 


• • • 


• • • 










+ 7 


+ 7 


- 21 


+ 7 


- 7 


+ 14 


- 7 


+ 1 










- 4 


- 7 


+ 9 


- 1 


+ 5 


- 5 


+ 1 












- 7 


+ 3 


+ 6 


- 2 


- 3 


+ 1 














+ 5 


- 1 


- 3 


• • • 


+ 1 
















+ 4 


+ 2 


- 4 


+ 1 


















- 1 


- 2 


+ 1 




















- 2 


+ 1 






















+ 1 
























































































































































































































































































































































































































































































































































































































































































































440 



[148 



148. 



MEMOIR ON THE RESULTANT OF A SYSTEM OF TWO 

EQUATIONS. 

[From the Philosophical Transactions of the Royal Society of London, vol cxLvn. for the 
year 1857, pp. 703—715. Received December 18, 1856,— Read January 8, 1857.] 

The Resultant of two equations such as 

(a, 6, ...50?, y)"* = 0, 
(p, q, ...5a?, y)'*=0, 

is, it is well known, a function homogeneous in regard to the coefficients of each 
equation separately, viz. of the degree n in regard to the coefficients (a, 6, ...) of 
the first equation, and of the degree m in regard to the coefficients (jd, g, ••) of 
the second equation; and it is natural to develope the resultant in the form 
AilP + A'-^.'P' + &c., where A, A\ &c. are the combinations (powers and products) of 
the degree n in the coefficients (a, 6, ...), P, P', &c. are the combinations of the 
degree m in the coefficients (p, g, ...), and k, Id, &c. are mere numerical coefficients. 
The object of the present memoir is to show how this may be conveniently eflfected, 
either by the method of symmetric functions, or from the known expression of the 
Resultant in the form of a determinant, and to exhibit the developed expressions for 
the resultant of two equations, the degrees of which do not exceed 4. With respect 
to the first method, the formula in its best form, or nearly so, is given in the 
Algebra of Meyer Hirsch, [for proper title see p. 417], and the application of it is very 
easy when the necessary tables are calculated: as to this, see my "Memoir on the 
Symmetric Functions of the Roots of an Equation "(^). But when the expression for the 
Resultant of two equations is to be calculated without the assistance of such tables, 
it is I think by far the most simple process to develope the determinant according 
to the second of the two methods. 

» Philosophical Transactiont, 1857, pp. 489—497, [147J. 



148] MEMOIR ON THE RESULTANT OF A SYSTEM OP TWO EQUATIONS. 441 

Consider first the method of symmetric ft^^ctioQS, and to fix the ideas, let the 
two equations be 

(a, b, c, cH^x, yY = 0, 

(jp, q, r "^x, yf = 0. 



Then writing 



so that 



(a, 6, c, d\l, zY = a(l -^ az)(l ^ fie)(l --yzl 



A = a + fi + y =(1), 
- f = <^7 - (I'). 



the Resultant is 



ip. 3. rSa, 1)' • ip, q, »*M 1)' • (p. q. rJi% lY, 

which is equal to 

r' + qr' (a + /5 + f) + pr' (a* + ^ + i*) + pqr (a?/5 + afi* + 0^ + fiy' + '^ + i'a) + Sk.; 

or adopting the notation for symmetric functions used in the memoir above referred 
to, this is 

{ r» 

{ + qr' (1) 

f+pr^ (2) 

X+q'r (P) 
j+pqr (21) 

1+9* (1*) 
'+ph- (2») 

+pq' (2P) 

{+p'q (2*1) 
l+P' (2*) . 



{ 



the law of which is best seen by dividing by r» and then writing 

? = [!]. f = [2]. 
and similarly, 

^ = [1']. f=[21].&c; 

56 



c. ir. 



442 



MEMOIR ON THE RESULTANT OF A SYSTEM OF TWO EQUATIONS. 



[148 



the expression would then become 

1 + [1] (1) + [2] (2) + [V] (V) + [21] (21) + [V] (P) + [2»] (2«) + [21»] (21«) + [2»1] (2n) + [2»] (2'), 

where the terms vdthin the [ ] and ( ) are simply all the partitions of the numbers 
1, 2, 3, 4, 5, 6, the greatest part being 2, and the greatest number of parts being 3. 
And in like manner in the general case we have all the partitions of the numbers 
1, 2, 3, ...mn, the greatest part being n, and the greatest number of parts being tn. 

The symmetric functions (1), (2), (1*), &c. are given in the Tables (6) of the 
Memoir on Symmetric Functions, but it is necessary to remark that in the Tables 
the first coefficient a is put equal to unity, and consequently that there is a power 
of the coefficient a to be restored as a factor: this is at once eflfected by the con- 
dition of homogeneity. And it is not by any means necessary to write down (as for 
clearness of explanation has been done) the preceding expression for the Resultant; 
any portion of it may be taken out directly from one of the Tables (6). For instance, 

the bracketed portion 

+ pqr (211 

+ ?' (1'). 

which corresponds to the partitions of the number 3, is to be taken out of the 
Table 111(6). as follows: a portion of this Table (consisting as it happens of consecutive 
lines and columns^ but this is not in general the case) is 



be 



+ 8 
-1 


-1 



= d 
(ii) 



if in this we omit thd sigh =, and in the outside line write for homogeneity ad 
instead of d, and in the outside column, first substituting q, p for 1, 2, then write 
for homogeneity pqr instead of pq, we have 



ad he 



pqr 



+ 3 
-1 


-1 

1 



viz. pjr X (+3ad — 16c) + g'(— lad), for the value of the portion in queistion; this is 
equivalent to 

vqr q^ ^^'''^^'^ 

, or as it may be mdre convenielitiy Writteti, 

be 

in which form it constitutes a part of the expression given in the sequel for the 
Resultant of the two functions in question; and similarly the remainder of the expres- 
sion is at once derived from the Tables (b) I. to VI. 



+ 3 -1 
-1 




148] 



MEMOIR ON THE RESULTANT OF A SYSTEM OF TWO EQUATIONS. 



443 



As a specimen of a mode of verification, it may be remarked that the Resultant 
qui invariant ought, when operated upon by the sum of the two operations, 

3a36 + 263o + c9rf and 2pdq + qdr, 

to give a result zero. The results of the two operations are originally obtained in the 
forms in the first and second columns, and the first column, and the second column, 
with all the signs reversed, are respectively equal to the third column, and conse- 
quently the sum of the first and second columns vanishes, as it ought to do. 





\: 



^d. 






\ 






o*. 






\ 



Next to explain the second method, viz. the calculation of the resultant from the 
expression in the form of a determinant. 



Taking the same example as before, the resultant is 

a, 6, c, d 
a, 6, c, dy 



56—2 



444 



MEMOIR ON THE BBSXTLTANT OF A SYSTEM OF TWO EQUATIONS. 



[148 



which may be developed in the form 



+ 12 . 345 } 

- 13 . 246 j 
+ 14 . 235 
+ 23 . 145 
- 15 . 234 

- 24 . 135 






where 12, 13, &c.* are the terms of 

( 



and 123, &c. are the terms of 



( 



+ 25 . 134 

+ 34 . 126 

-36 . 124} 

4-45 . 123} 

a, 6, c, d ) 

a, b, c, d 

p, q. r 



viz. 12 is the determinant formed with the first and second oolumns of the upper 
matrix, 123 is the determinant formed vdth the first, second and third columns of 
the lower matrix, and in like manner for the analogous symbols. These determinants 
must be first calculated, and the remainder of the calculation may then be arranged 
as follows: — 



\: 



c^. 



\ 



\ 



r X < 




J 



> ^ < 



\ 



«v. 



«♦ 



148] 



MEMOIR ON THE RESULTANT OF A SYSTEM OF TWO EQUATIONS. 



445 



where it is to be observed that the figures in the squares of the third column are 
obtained from those in the corresponding squares of the first and second columns by 
the ordinary rule for the multiplication of determinants, — ^taking care to multiply the 
dexter lines (ie. lines in the direction \) of the first square by the sinister lines 
(i.e. lines in the direction/) of the second square in order to obtain the sinister lines 
of the third square. Thus, for instance, the figures in the square 




are obtained as follows, viz. the first sinister line (-h 3, — 1) by 

(-1, +l)(-2, +1)= 2 + l = + 3, 
(-1, +1)(+1, 0) = -l + = -l, 

and the second sinister line (— 1, 0) by 

(0, -l)(-2, +1) = 0-.1=-1, 

(0, -l)(^-l, o) = o + o= 0. 

I have calculated the determinants required for the calculation, by the preceding 
process, of the Resultant of two quartic equations, and have indeed used them for 
the verification of the expression as found by the method of symmetric functions; as 
the determinants in question are useful for other purposes, I think the vahies are 
worth preserving. 



Table of the Determinants of the Matrices, 



and 



( 



( 









a, 


6, 


c, 


d, e ) 






a, 


6, 


c, 


d, 


e 




a, 


6, 


c, 


d, 


e 




a, 


6, 


c, 


d, 


«, 






f 






P» 


tf» 


r, 


8, t ) 






P> 


?» 


r. 


s, 


t 




P* 


q> 


r, 


«. 


t 




P> 


9» 


r, 


s, 


t 







HEHOIB OH THE RESULTANT OF A SYSTEH OF TWO EQUATIONS. 



[148 



arranged in the form adapted for the calculation of the Resultant of the two qiuutic 
equations (a, b, c, d, e^x, yf = 0, and (p, q, r, s, t'Sx, y)* = 0, viz. 





148] MEMOIR ON THE BE8ULTANT OF A SYSTEM OF TWO EQUATIONS. 447 




448 MEMOIR ON THE RESULTANT OF A SYSTEM OF TWO EQUATIONS. [l48 




-/^■^ 



■r % 



/\ 




148] MEMOIR ON THE RE-SHLTANT OF A SV8TGM OF TWO EQUATIONS. 449 





The Tables of the Resultants of two uquatious which I have calculated are as 



450 



MEMOIR ON THE RESULTANT OF A SYSTEM OF TWO EQUATIONS. 



[148 



Table (2, 2). 

Resultant of 

(a, 6, c\x, yY, 



Table (3, 2). 

Resultant of 
(a, 6, c d^x, yy, 
(/>, q. r "^x, yy. 



Table (4, 2). 
Resultant of 
(a, 6, c, d, e^x, yy, 
(p, 9, r^x, yy. 





\ 



\/ 








148] MEMOIR ON THE RESULTANT OF A SYSTEM OF TWO EQUATIONS. 

Table (*. 3). 

Resultant of 

(a, 6, c, d. e$x, y)*, 

ip, q, r, sXx, yf. 



Table (3, 3). 


Resultant of 


(a, b. c. d5», sr. 


(^5, r, .5., y)-. 


yA 


'0 


'>\<> 








MEMOIR ON THE BE8ULTANT OF A SYSTEM OF TWO EQUATIONS. 

Table (4, 4), 

RfsiiUiint of 
(a, 6, c, d, e^x; yY. 
(p. q, r. s, l\x. yy. 



[148 




148] MEMOIR ON THE RESULTANT OF A SYSTEM OF TWO EQUATIONS. 453 





454 



[149 



149. 



ON THE SYMMETRIC FUNCTIONS OF THE ROOTS OF CERTAIN 

SYSTEMS OF TWO EQUATIONS. 



[From the Philosophical Transactions of the Royal Society of London, vol. CXLVII. for the 
year 1857, pp. 717 — 726. Received December 18, 1856, — Read January 8, 1857.] 

Suppose in general that ^ = 0, -^ = 0, &c. denote a system of (n — 1) equations 
between the n variables {x, y, z, ...), where the functions <f>, y^,' &c. are quantics ^e. 
rational and integral homogeneous functions) of the variable& Any values {xi, yi, Zi,...) 
satisf}dng the equations, are said to constitute a set of roots of the system; the roots 
of the same set are, it is clear, only determinate to a common factor pris, Le. only 
the ratios inter se and not the absolute magnitudes of the roots of a set are deter- 
minate. The number of sets, or the degree of the system, is equal to the product 
of the degrees of the component equations. Imagine a function of the roots which 
remains unaltered when any two sets (o^i, yi, Zi, ...) and (^, y,, z^, ...) are interchanged 
(that is, when Xi and x^, yi and y,, &c. are simultaneously interchanged), and which is 
besides homogeneous of the same degree as regards each entire set of roots, although 
not of necessity homogeneous as regards the different roots of the same set; thus, 
for example, if the sets are (xi, yi), (x^, y,), then the functions XyX^, XtJ/^ + x^i, y^y^ 
are each of them of the form in question; but the first and third of these functions, 
although homogeneous of the first degree in regard to each entire set, are not homo- 
geneous as regards the two variables of each set. A function of the above-mentioned 
form may, for shortness, be termed a symmetric function of the roots; such function 
(disregarding an arbitrary factor depending on the common factors which enter implicitly 
into the different sets of roots) will be a rational and integral function of the coefficients 
of the equations, i.e. any symmetric function of the roots may be considered as a 
rational and integral function of the coefficients. The general process for the investi- 
gation of such expression for a symmetric function of the roots is indicated in Pro- 
fessor Schlafli's Memoir, "Ueber die Resultante eines Systemes mehrerer algebraischer 



149] ON THE SYMMETRIC FUNCTIONS OF THE ROOTS, &C. 455 

Gleichungen," Vienna Transactions, t. iv. (1852). The process is as follows: — Suppose 
that we know the resultant of a system of equations, one or more of them being 
linear; then if ^ = be the linear equation or one of the linear equations of the 
system, the resultant will be of the form 0i^..., where <^i, ^, &c. are what the 
function <}> becomes upon substituting therein the different sets (a?i, yi, -2^1 ...)> (^i ya* -^^a**-) 
of the remaining (n — 1) equations -^ = 0, X ~ ^' ^* > comparing such expression with 
the given value of the resultant, we have expressed in terms of the coefficients of the 
functions y^, x* ^j certain symmetric functions which may be caHed the fundamental 
symmetric functions of the roots of the system -^ = 0, % = 0, &c. ; these are in fact 
the symmetric functions of the first degree in respect to each set of roots. By the 
aid of these fundamental symmetric functions, the other symmetric functions of the 
roots of the system -^ = 0, x ~ ^» ^- ^^7 ^ expressed in terms of the coefficients, 
and then combining with these equations a non-linear equation * = 0, the resultant 
of the system 4> = 0, ^^ = 0, x^^> ^- ^^^ ^ what the function *i*i . . . becomes, upon 
substituting therein for the different symmetric functions of the roots of the system 
•^ = 0, x = 0, &c. the expressions for these functions in terms of the coefficients. We 
thus pass from the resultant of a system ^ = 0, '^ = 0, x = 0, &c., to that of a system 
4> = 0, V^ = 0, X " ^> ^» ^^ which the linear function <f> is replaced by the non-linear 
function *. By what haa preceded, the symmetric functions of the roots of a system 
of (n — 1) equations depend on the resultant of the system obtained by combining the 
(n—l) equations with an arbitrary linear equation ; and moreover, the resultant of any 
system of ri equations depends ultimately upon the resultant of a system of the same 
number of equations, all except one being linear; but in this case the linear equations 
determine the ratios of the variables or (disregarding a common factor) the values of 
the variables, and by substituting these values in the remaining equation we have the 
resultant of the system. The process leads, therefore, to the expressions for the 
synmietric functions of the roots of any system of (w— 1) equations, and also to the 
expression for the resultant of any system of n equations. Professor Schlafli discusses 
in the general case the problem of showing how the expressions for the fundamental 
symmetric functions lead to those of the other symmetric functions, but it is not 
necessary to speak further of this portion of his investigations. The object of the 
present Memoir is to apply the process to two particular cases, viz. I propose to 
obtain thereby the expressions for the simplest symmetric functions (after the funda- 
mental ones) of the following systems of two ternary equations; that is, first, a linear 
equation and a quadric equation ; and secondly, a linear equation and a cubic 
equation. 

First, consider the two equations 

(«, 6, c, /, g, h^x, y, zy = 0, 
(a, /8, yj^x, y, z) = 0, 

and join to these the arbitrary linear equation 

(f , V> ?$^, y, ^) = 0, 



456 



ON THE SYMMETRIC FUNCTIONS OF THE ROOTS 



[149 



then the two linear equations give 

and substituting in the quadratic equation, we have for the resultant of the three 
equations, 

(a, 6, c,/ g, hJi^^^ryr,, 7? - «?, a^-/9f)^ = 0, 
which may be represtented by 

(a, b, c, f, g, h$f 17, f)> = 0, 
where the coefficients are given by means of the Table. 





a 


b 


c 


f 
-2)8y 


9, 


h 




a=> 


+ >* 


+ fi' 






1 (a 


b = 


+ y' 




+ a' 




-2ya 




(-J*) 


c = 


+ /3» 


+ o» 





-a* 


+ afi 




(D 


f = 


-Py 


2(i»0 


g = 




-ya 




+ a)8 


-P" 


+ ;3y 


2«0 


h= 






-aP 


+ ya 


^Py 


-y" 


2(^) 



viz. a = 67* + C)8* - 2/597, &c. 

But if the roots of the given system are 

then the resultant of the three equations will be 
and comparing the two expressions, we have 



a = 


'X1X2 


i 


b = 


yiy« 


» 


c = 


-2^1-8^2 


i 


2f = 


yi^2 


+ y^i> 


2g = 


ZlX^ 


^-z^i. 


2h = 


^iVi 


+ arjyi, 



which are the expressions for the six fundamental symmetric functions, or symmetric 
functions of the first degree in each set, of the roots of the given system. 

By forming the powers and products of the second order a', ab, &c., we obtain 
linear relations between the symmetric fimctions of the second degree in respect to 
each set of roots. The number of equations is precisely equal to that of the 



149] 



OF CERTAIN SYSTEMS OF TWO EQUATIONS. 



457 



symmetric functions of the form in question, and the solution of the linear equations 

gives — 

a* = x^x^, 

b* = yi V. 

he = ViZ^^t , 
ca = ZiXiZ^ , 
ab = x^iXttf,, 



4f» 
4g» 
4h» 



2bc = 
2ca = 
2ab = 

2af = 
2bg = 
2ch = 






4gh - 2af = a^»y^j + aJj^yiiTi, 
4hf - 2bg = yi«-e:jic, + yj^iTja?!, 
4fg - 2ch = -gTi^j?,^, + z^Xjy^ , 

2bf = yi»y^, + ya^yi-gr^ , 
2cg = V^,a?a + 2rj9iria?i , 
2ah = iCi'^j + a;i*a?iyi, 

2cf = Vya^i + ^a^^i , 
2ag = Xi^^ + x^ZiX^ , 
2bh = yi«aj,y8 + ya«a7iyi. 

Proceeding next to the powers and products of the third order a', a'b, &c., the 
total number of linear relations between the symmetric functions of the third degree 
in respect to each set of roots exceeds by unity the number of the symmetric functions 
of the form in question; in fact the expressions for abc, aP, bg*, ch', fgh, contain, 
not five, but only four symmetric functions of the roots; for we have 

abc = x^^Zy^ . x^iZ^y 

4af» = {xjy^x^i + x^^x^z^) + ^lyiZiX^^^, 
4bg» = (j/iZih/^^^ + j/j^aVi^O + ix^y^z^x^^^, 
4ch* = {z^x^z^^ + z^cfzTjjf) + ^i^xZ^x^^^, 
8fgh = {x^^x^i + x^}x^z^ ) ' 

+ {yxZxV^i + y%z^xx^) - + 2a^yi-^i^a^j, 

+ {Z^X^Z^} + Z^^ZtJI^ ) , 



C. II. 



58 



458 ON THE SYMMETBIO FUNCTIONS OF THB ROOTS [149 

and consequently the quantities a, b, c, f, g, h, are not independent, but are connected 

by the equation 

abc - aP - bg« - ch« + 2fgh = 0, 

an equation, which is in fact verified by the foregoing values of a, &c. in terms of 
the coeflScients of the given sjrstem. 

The expressions for the symmetric functions of the third degree considered as 
Hinctions of a, b, c, f, g, h, are consequently not absolutely determinate, but they may 
be modified by the addition of the term \ (abc — af* — bg* — ch' + 2fgh), where \ is an 
indeterminate numerical coefficient. 

The simplest expressions are those obtained by disregarding the preceding equation 
for fgh, and the entire sjrstem then becomes : 

at ^_ /M S/ft 3 
^— U/i U/] , 



b' =»yiV» 

b«c = yi'^iy,"^, , 
c^a ^ JB| SC\Z^ cc^ f 

bc» = VxZ^^f , 
ca" ^ZiX^z^, 
ab« = xijf^^x^^, 

abc s= ohyiZiX^tSt^ 

2a»f = x^^yiz^^ + x^y^xO^. 
2b«g= y^z^x^}-\-y}z^^^, 
2c»h = Zx^x^^^ + z^x^iZx\ 

2a fi ^~ x^ z^fjc^ "T x^ i9]M>| , 

2b«h = yx^x^^^ + y2Si>{yx\ 
2c»f = Vy^,« + z^y,z,\ 

2a'h = x^z}x^ + x^z^Xy , 
2b«f = yi»a:,«yj + y,»a^«yi> 
2c*g = z^%^z^ + z^h/^^Zx , 

2bcf = yx^z^^f + y}z^xz^ , 
2cag= z^XiZ^ +z^XiZiX^, 
2abh= x^ix^^ ^ x^^iOxyx\ 

2bcg = yxz^x^^^ + y^ix^xZx , 
2cah = ZiX^x^^2 + z^x^iZ^ , 
2abf = x^j^x^^t + x^^xiy^z^ , 



149] OF CERTAIN SYSTEMS OF TWO EQUATIONS. 459 







2bch= 


VxZ^^H^t + yfz^^xZx , 






2caf = 


z^x^x^^^ + zix^yyz^ , 






2abg= 


x^%x^^^ + iJc^y^iyxZx. 




4af» 


-2abc = 


a?iy,V^j+a?j^aV^, 




4bg» 


-2abc = 


yiV^aVs + yV^q^y,, 




4ch« 


-2abc = 


Zx^y}Z% + Z^}\)^Z^ , 




4bf» 


-2b*c = 


yxy^%-^yiyxz^. 




4cg» 


-2c«a = 


z^z^i + z}z^x^ , 




4a.h« 


-2a«b = 


x^x^^ + x}x^^. 




4cP 


-2bc« = 


Zx^%z^ + z^yx^z^ , 




4ag». 


-2ca« = 


x^z^x^ + a^ V^ » 




4bh«. 


-2ab» = 


yx*^9%-^yMY> 




4agh- 


-2a2f = 


x^^x^^^ + x^x^y^z^. 




4bhf- 


- 2b«g = 


yi*x^^^ -h yj'x^iZi, 




4cfg - 


-2c«h = 


Z^^^% + ZfXTjj^Zy^ , 




4bgh' 


- 2abf = 


yxZx^%y%-^ y^z^y^. 




4chf 


- 2bcg = 


z^x^iz^ + z}x^^z^ , 




4afg > 


-2cah = 


O^xZix^-^ O^^^^x. 




4cgh • 


-2acf = 


yxz^xfz^ -h y^ix^Zx , 




4ahf 


-2bag = 


ZxX^}x^-\- z^^y^x^. 




4bfg 


- 2cbh = 


i^xUxZ^y^^r x^fz^y^. 


SPg- 


- 4chf . 


-2bcg = 


z^x^} + z^m^. 


8g«h. 


-4a% - 


-2cah = 


x^y^} + xiy^z^. 


8h«f- 


- 4bgh • 


- 2abf = 


yxz^i + y^z^x^. 


8fg»- 


-4chg 


-2acf = 


V^'ya + V^'yi, 


8gh«. 


-4afh- 


- 2bag = 


a^yfz^^- x^y^z^, 


8hf«. 


-4bgf . 


- 2cbh = 


y^zix^-^-yiz^Xx, 




8f» ■ 


- 6bcf = 


yxzi + y,V, 




8g» 


-6cag = 


z^x{ + z}x^y 




8h» . 


- 6abh = 


«i V + «a V- 



Secondly, consider the system of equations 

(a, 6, c, /, flf, A, i, j, fc, fja?, y, ^)» = 0, 
(a, /9, 7]^a?, y, -^) = 0, 



58—2 



460 



ON THE SYMMETRIC FUNCTIONS OF THE ROOTS 



[149 



where the cubic function written at full length is 
Joining to the system the linear equation 

(f I % ?$«. y, ^) = 0, 

the linear equations give 
and the resultant is 

which may be represented by 

(a, b, c, f, g. h, i, j, k, l$f , 17, ?)> = 0, 
where the coefficients a, b, &c. are given by means of the Table : — 



a 



f 



a = 
b = 
c = 






-3-ya» 


-3ai8» 


+ 3i8V 


+ 87*0 


+ 3a')3 
-2o/3y 




f 


f = 

g = 
b = 






-/a 

+ /3V 


-a* 
-laJfiy 


-2a)3y 


+ 2ya' 
+ 20/3' 
+ 2)8y' 

-2o»/3 
-2y*a 


3« 
3f«i, 


• 

1 = 

• 

J = 
k= 


-A 

-o»6 


+ a» 
+ 2a^y 


+ 2o/8y 


+ 2aj8y 


-a/3» 
+ ya' 




-ya« 
+ /8y' 


3{^ 1 
3^ 

■ 


1 = 




-y«« 


-«y3» 


-H 


+ a»/3 


+y3'y 


+ 7*0 




6M; 



VIZ. 



a = 67* - ciS* - 3/i87» + 3ti8«7, &c. 



But if the roots of the given system are 

then the resultant of the three equations may also be represented by 

and comparing with the former expression, we find : 

a ^^ X\X^-^y 
b = yiy^a^j, 

C s= Z\Z^^^ 



149] OF CERTAIN SYSTEMS OF TWO EQUATIONS. 461 

3f = yiy^t + y^^x + y^iz^. 
3g = ZiZ^x^ + z^z^ + z^iXi, 

3i = y^z^t + ya«r^i + y^^z^, 

3j = -e^iiTa^s + Z^X>^ + ZgXiX^y 

3k = ay^^, + iC8yayi+ a?,yiyj, 

But there is in the present case a relation independent of the quantities a, &c., viz. 
we have (a, )8, 7$iCi, yu ^i) = 0, (a, )8, y^x^, y,. -8^,) = 0, (a, $, 7$j^, y„ z,) = 0, and 
thence eliminating the coefficients (a, /8, 7), we find 

By forming the powers and products of the second degree a', ab, Ac, we obtain 55 
equations between the symmetric functions of the second degree in each set of roots. 
But we have V = = a symmetric ftmction of the roots, and thus the entire number 
of linear relations is 56, and this is in fact the number of the synmietric functions 
of the second degree in each set. I use for shortness the sign S to denote the sum 
of the distinct terms obtained by permuting the different sets of roots, so that the 
equations for the fundamental symmetric functions are — 

a = ^^2^> 
b= y,y^„ 

C = ZiZ^Zit 

3f = S yij/j^s, 
3g=S-?i^2a?8, 
Sh = SxjX^i, 

3i =SyiZ^z^, 
3j =SziX^3, 
3k = S x^^t, 

61 =Sx{y^^\ 

then the complete system of expressions for the symmetric functions of the second 

order is as follows, viz. 

a« = x^x^x^, 

b» = yWyz\ 

c« = z^^z^H,\ 
be = yiz^y^^^t, 

ca -^ Zi^XiZ^flc^ZfjX^f 
ab= Xiy^x^^^t, 



462 ON THE SYMMBTBIC FUNCTIONS OF THE BOOTS [l49 

3af =^ S XitfiX^^^Xt, 

3ch = S siiXiZ^x^^z^, 
3bf =iSyi»y,»y,^„ 

3cg=iS-fxV'8^s«8, 

3ah = Sx^x^x^i, 

3cf = S yiz,y^ztz^\ 
Ssig =^ S z,x,z,fc^\ 
Sbh = S x^ix^^^\ 

3ai - S x^^z^^z^j^, 
3bj ^Sj/iz^x^^^^, 
3ck = S ZiXiy^^iZs, 

3bi = 8 yi^^^tz^, 
3cj ^Szi^z^z^, 
3ak = jS ari«a?3^jfl^„ 

3ci ^SyiZiZ^z^, 
3aj =SziXiXi^Xi\ 
ahk = Sx^^^y^\ 

6al =Sxi^x^iZ^^, 
6bl = iS yi*y^,a^„ 
6cl = S Zi^z^x^^z^, 

9f« - 6bi = S y,«y,V, 

9g» -6cj =Sz,W^\ 
9h» - 6ak = iS Xx"a:i«y,«, 

9i'» - 6cf = iS yl«-^a«-^,^ 
9j« -6ag = S^,»a:,%», 
9k« - 6bh = S a?,«y,V. 

9fg - 3ck = iS Xjy^^^s\ 
9gh-3ai ^Sy^z^z^^^\ 
9hf-3bj ^SziXjX^^i^ 



149] OP CERTAIN SYSTEMS OF TWO EQUATIONS. 463 

9jk - 3af = Sxi^x^^j^i, 
9ki - 3bg = fif y^y^^z^^, 
9ij — 3ch = S z^z^xi^^^, 

9f i - 3bc = S y,^y,z^,\ 
9gj — 8ca = S z^z^^x^y 
9hk~ Sab = fif x^^x^^^, 

3 ( Q + gk + hi - 1«) = Sx^,z^x^^^, 

3(2fj -gk-hi + l«) = fif^iy,a?4^^,«, 
3(2gk -hi -fj +\*) = 8y,z,y,z^,\ 
3(2hi -fj -gk + l«) = flf^ia?,^^,y,«, 

3(6fl -3ki-.bg) =flfay^^,V, 
3(6gl -3ij-ch) =8y,z,z,W, 
3(6hl -3jk-af) =^Sz^x^x^^^\ 

3(6il -3fg-ck) =SziX^^%\ 
3(6jl -3gh-ai) ^Sx^,zW. 
3(6kl-3hf-bj) ^Sy,z,x,%\ 

6(~fj-gk-hi + 41«) = fifa:i»y,V. 

As an instance of the application of the formulae, let it be required to eliminate 
the variables from the three equations, 

(a, 6, c, /, g, h, i, j, A, i$a;, y, ^)» = 0, 
(a\ b\ c\ /, ^, K Jx, y, zf = 0, 

(a, yS, 7 $a;, y, 2^) = 0. 

This may be done in two different ways; first, representing the roots of the linear 
equation and the quadric equation by (a?i, yi, ^,), (a^, y,, z^, the resultant will be 

(tt, ...$071, yi, ^,)».(a,...Ja;i, y,, -erj)*, 
which is equal to 

a* aJj'^Tj' + &C., 

where the symmetric functions x^x^, &c. are given by the formulae a'* = ajj'a?,*, &c., 
in which, since the coefficients of the quadratic equation are {a\ h\ c\ f\ gf^ h'\ 
I have written a' instead of a. Next, if the roots of the linear equation and the cubic 
equation are represented by {xi, yi, z^, (a?,, yj, z^, (x^, y,, -?,), then the resultant 
will be 

(tt', ...$da, yi. ^,)'.(a',...$d:2, ya, z^y(a\ ...^x^, y„ -^j^, 



464 ON THE SYMMETRIC FUNCTIONS OF THE ROOTS &C. [149 

which is equal to 

a'* ayfxfx^ + &c., 

the symmetric functions x^^x^x^, &a being given by the formulaB d?=^x^x^x^, &c. The 
expression for the Resultant is in each case of the right degree, viz. of the degrees 
6, 3, 2, in the coefficients of the linear, the quadric, and the cubic equations respec- 
tively: the two expressions, therefore, can only differ by a numerical factor, which 
might be determined without difficulty. The third expression for the resultant, viz. 

(where (a?,, yi, Zi\,..(x^, y^, z^ are the roots of the cubic and quadratic equations) 
compared with the foregoing value, leads to expressions for the fundamental sjrmmetric 
functions of the cubic and quadratic equations, and thence to expressions for the other 
symmetric functions of these two equations; but it would be difficult to obtain the 
actually developed values even of the fundamental symmetric functiona I hope to 
return to the subject, and consider in a general point of view the question of the 
formation of the expressions for the other symmetric functions by means of the ex- 
pressions for the fundamental symmetric functiona 



150] 



465 



150, 



A MEMOIR ON THE CONDITIONS FOR THE EXISTENCE OF 
GIVEN SYSTEMS OF EQUALITIES AMONG THE ROOTS OF 
AN EQUATION. 



[From the Philosophical Transactions of the Royal Society of London, vol. CXLVII. for 
the year 1857, pp. 727 — 731. Received December 18, 1856, — Read January 8, 1857.] 

It is well known that there is a symmetric function of the roots of an equation, 
viz. the product of the squares of the differences of the roots, which vanishes when any 
two roots are put equal to each other, and that consequently such function expressed in 
terms of the coefficients and equated to zero, gives the condition for the existence of a 
pair of equal roots. And it was remarked long ago by Professor Sylvester, in some of 
his earlier papers in the Philosophical Magazine, that the like method could be applied 
to finding the conditions for the existence of other systems of equalities among the roots, 
viz. that it was possible to form symmetric functions, each of them a sum of terms 
containing the product of a certain number of the differences of the roots, and such that 
the entire function might vanish for the particular system of equalities in question ; 
and that such functions expressed in terms of the coefficients and equated to zero would 
give the required conditions. The object of the present memoir is to extend this theory 
and render it exhaustive, by showing how to form a series of types of all the different 
functions which vanish for one or more systems of equalities among the roots; and in 
particular to obtain by the method distinctive conditions for all the different systems of 
equalities between the roots of a quartic or a quintic equation, viz. for each system con- 
ditions which are satisfied for the particular system, and are not satisfied for any other 
systems, except, of course, the more special systems included in the particular system. 
The question of finding the conditions for any particular system of equalities is essen- 
tially an indeterminate one, for given any set of functions which vanish, a function 
syzygetically connected with these will also vanish; the discussion of the nature of the 
C. II. 59 



466 ON THE CONDITIONS FOR THE EXISTENCE OP GIVEN [150 

syzygetic relations between the diflFerent functions which vanish for any particular 
system of equalities, and of the order of the system composed of the several conditions 
for the particular system of equalities, does not enter into the plan of the present 
memoir. I have referred here to the indeterminateness of the question for the sake of 
the remark that I have availed myself thereof, to express by means of invariants or 
covariants the different systems of conditions obtained in the sequel of the memoir; the 
expressions of the different invariants and covariants referred to are given in my 'Second 
Memoir upon Quantics,' Philosophical TranscLCtions, vol. CXLVI. (1856), [141]. 

1. Suppose, to fix the ideas, that the equation is one of the fifth order, and call 
the roots a, 13, % S, e. Write 12 = 2<^(a-i8)Ml2.13 = 2<^(a-i8y(a-7)"», 12.34 = 
2^ (a — i8)'(7 — 8)**, &c., where ^ is an arbitrary function and i, m, &a are positive integers. 
It is hardly necessary to remark that similar types, such as 12, 13, 45, &c., or as 12.13 
and 23.25, &c., denote identically the same sums. Two types, such as 12.13 and 
14.15.23.24.25.34.35.45, may be said to be complementary to each other. A par- 
ticular product (a — yS)(7 — 8) does or does not enter as a term (or factor of a term) 
in one of the above-mentioned sums, according as the type 12.34 of the product, or 
some similar type, does or does not form part of the type of the sum; for instance, the 
product (a — i8)(7 — S) is a term (or factor of a term) of each of the sums 12.34, 
13.45.24, &c., but not of the sums 12.13.14.15, &a 

2. If, now, we establish any equalities between the roots, e.g. ci = l3, 7 = 8, the 
effect will be to reduce certain of the sums to zero, and it is easy to find in what 
cases this happens. The sum will vanish if each term contains one or both of the &ctors 
^"ffi 7— Si i.e. if there is no term the complementary of which contains the product 
(a — yS) (7 — 8), or what is the same thing, whenever the complementary type does not con- 
tain as part of it, a type such as 12.34. Thus for the sum 14.15.24.25.34.35.45, 
the complementary type is 12.13.23, which does not contain any type such as 12.34, 
i. e. the sum 14.15.24.25.34.35.45 vanishes for a = y3, 7 = 8. It is of course clear 
that it also vanishes for a = yS = e, 7 = 8 or a = yS = 7 = 8, &a, which are included in 
^ = fi» 7=8. But the like reasoning shows, and it is important to notice, that the 
sum in question does not vanish for a = ^ = y: and of course it does not vanish for 
a = ^. Hence the vanishing of the sum 14.15.24.25.34.35.45 is characteristic of the 
system a = yS, 7=8. A system of roots a, yS, 7, 8, e may be denoted by 11111; but 
if a = y3, then the system may be denoted by 2111, or if a = i8, 7 = 8, by 221, and 
so on. We may then say that the sum 14.15.24.25.34.35.45 does not vanish for 
2111, vanishes for 221, does not vanish for 311, vanishes for 32, 41, 5. 

3. For the purpose of obtaining the entire system of results it is only necessary to 
form Tables, such as the annexed Tables, the meaning of which is sufficiently explained 
by what precedes: the mark (x) set against a type denotes that the sum represented 
by the complementary type vanishes, the mark (o) that the complementary type does 
not vanish, for the system of roots denoted by the symbol at the top or bottom of the 
column; the complementary type is given in the same horizontal line with the original 
type. It will be noticed that the right-hand columns do not extend to the foot of the 
Table ; the reason of this of course is, to avoid a repetition of the same type. Some of 



BTSTEH8 OF EQUAUTIES AUONQ THE ROOTS OF AN EQUATION. 



467 



150] 

the types at the foot of the Tables are complemeDtaiy to themselves, but I have, not- 
withstauding this, given the complementary type in the form under which it naturally 
presents itself. 

4. The Tahles are: 

Table for the equal Boots of a Quartic. 






211 


22 


31 


4 


14 . 23 . 24 . 34 


o 


o 


■s' 


o 


14 . 23 . S4 . 34 


o 


o 


o 


X 


14 . 23 . 24 . 34 


o 


o 


o 


» 


13 . 14 . 23 . 24 


a 


o 


X 


X 


"IT 23 . 24 . 34 





— 


— 


— 


84 14 . 23 . 34 


sn 


22 


31 


4 


23 14 . 24 . 34 











Table for the equal Boots of a Quintia 



Sill 


221 


311 


32 


*1 


fi 


~^ 






T 




7 


o 


X 


X 


X 


X X 


o 


X 


X 


X 


X X 


o 


o 


X 


X 


X ly 


o 


X 


x 


X 


X ix 


o 


o 


X 


X 


xlx 


o 





X 


X 


X jx 


o 


y. 


o 


y 


X |x 


o 


X 


X 


X 


X X 


o 


o 


a 


X 


X > 


o 


o 


X 


X 


X X 




o 


X 


X 


X 


X 




O 


X 


X 


X 


X 










o 


X 


X 


o 


o 


o 


X 


X 


X 


o 


o 





X 


X 


X 


o 


o 


o 


X 


X 


X 


o 


o 





X 


X 


X 


o 


a 


o 


o 


X 


X 


o 


o 


X 


X 


X 


^ 


Bill 


ssT 


311 


32 


41 


7 






2111 


2« 


51 


32 


41 


b 


35 . 46 




o 





~~^ 


~^ 


a 


35 . 45 




a 











X 


35 . 45 


o 














X 


35 . 45 


o 


a 








X 


X 


36 . 46 


o 


o 


a 








y 


34 . 35 


o 


o 








X 


X 


35 . 46 


o 


o 








X 


X 


35 . 46 


o 


o 





X 


X 


X 


35 . 45 


o 


o 





X 





X 


36 . 46 


o 


o 





X 


X 


X 


36 . 46 


o 








X 


X 


X 


36 . 46 


o 


o 








X 


X 


34 . 45 


o 


o 








X 


X 


34 . 36 


o 


o 


X 


X 


X 


X 


35 . 45 
35 . 45 














2111 


221 


311 


32 


41 


b 


36 . 45 


' 






35 . 46 








34 . 36 








26 . 36 















The two Tables enable the discussion of the theory of the equal roots of a quartic or 
quintic equation: first for the quartic: 

5. In order that a quartic may have a pair of equal roots, or what is the same 
thing, that the system of roots may be of the form 211, the type to be considered is 
12.13.14.23.24.34; 

59—2 



468 ON THE CONDITIONS FOR THE EXISTENCE OF GIVEN [l50 

this of course gives as the function to be equated to zero, the discriminant of the 
quartic. 

6. In order that there may be two pairs of equal roots, or that the system may 
be of the form 22, the simplest type to be considered is 

14.24.34; 
this gives the function 

which being a covariant of the degree 3 in the coefficients and the degree 6 in the 
variables, can only be the cubicovariant of the quartic. 

7. In order that the quartic may have three equal roots, or that the system of 
roots may be of the form 31, we may consider the type 

13.14.23.24, 

and we obtain thence the two functions 

2(a-7)(a-S)(iS-7)(^-S), 
2(a-7)«(a-8)(^-7)(/3-8)', 

which being respectively invariants of the degrees 2 and 3, are of course the quadrin- 

variant and the cubinvariant of the quartic. If we had considered the apparently more 

simple type 

12.34, 
this gives the function 

which is the quadrivariant, but the cubinvariant is not included under the type in 
question. 

8. Finally, if the roots are all equal, or the system of roots is of the form 4, then 
the simplest type is 

12; 
and this gives the function 

a covariant of the degree 2 in the coefficients and the degree 4 in the variables ; this is 
of course the Hessian of the quartic. 

Considering next the case of the quintic: 

9. In order that a quintic may have a pair of equal roots, or what is the same 
thing, that the system of roots may be of the form 2111, the type to be considered is 

12.13.14.15.23.24.25.34.35.45; 

this of course gives as the function to be equated to zero, the discriminant of the 
quintic. 



150] SYSTEMS OP EQUALITIES AMONG THE ROOTS OF EQUATION. 469 

10. In order that the quintic may have two pairs of equal roots, or that the 
system of roots may be 221, the simplest type to be considered is 

14.15.24.26.34.35.45; 

a type which gives the function 

2 (a - S)(a - e)(^ - 8)(yS - e)(7 - S)(7 - e)(B - €)« (a; - ay)' (x - ySy)« (x - 7y)». 

This is a covariant of the degree 5 in the coefiBcients and of the degree 9 in the variables ; 
but it appears from the memoir above referred to, that there is not any irreducible 
covariant of the form in question; such covariant must be a sum of the products 
(No. 13)(No. 20), (No. 13)(No. 14)», (No. 15)(No. 16) (the numbers refer to the Cova- 
riant Tables given in the memoir), each multiplied by a merely numerical coefficient. 
These numerical coefficients may be determined by the consideration that there being 
two pairs of equal roots, we may by a linear transformation make these roots 0, 0, oo , oo , 
or what is the same thing, we may write a = 6 = e=/=0, the covariant must then 
vanish identically. The coefficients are thus found to be 1, — 4, 50, and we have for a 
covariant vanishing in the case of two pairs of equal roots, 

1 (No. 13)(No. 20) 
- 4 (No. 13)(No. Uy 
+ 50 (No. 15)(No. 16) 
[or in the new notation AH — ^AB* + 50CD]. 

In fact, writing a = 6 = e=/=0, and rejecting, where it occurs, a factor ic'y*, the several 
covariants become functions of ex, di/; and putting, for shortness, x, y instead of ex, dy, 
the equation to be verified is 

1 . 10(a: + y)(6a:* + Sa^y + 28«y + Sxy^ + 6y*) 
- 4i.lO{x'\-y){Za? + %xy + Zr/'y 
+ 50(6a;* + &ry + 6y»)(a^ + ic'y + a?y« + y») = ; 

and dividing out by {x + y) and reducing, the equation is at once seen to be identically 
true. 

11. In order that the quintic may have three equal roots, or that the system 
of roots may be of the form 311, the simplest type to be considered is 

12.13.23.45; 

this gives the function 

2(a-yS)>(i8«7)'(7-«)'(S-«)*, 

which being an invariant, and being of the fourth degree in the coefficients, must be 
the quartinvariant of the quintic [that is No. 19, = 0\ The same type gives also the 
function 

2(a-^)«(i8-7)«(7-a)«(S-.6)«(a:-.Sy)»(a:-.6y)», 



470 ON THE CONDITIONS FOR THE EXISTENCE OF GIVEN SYSTEMS, &C. [150 

which is a covariant of the degree 4 in the coefficients and the degree 4 in the 
variables; and it must vanish when a = 6 = c = 0, this can only be the covariant 

3 (No. 20) -2 (No. 14)>, [=3ir-25«], 
which it is clear vanishes as required. 

12. In order that the quintic may have three equal roots and two equal roots, 
or that the system of roots may be of the form 32, the simplest type to be con- 
sidered is 

12.13.14.15, 

which gives the function 

2(a-^)(a-7)(a-S)(a-e)(a;-ySy)«(^-7y)*(a?-Sy)«(^-6y)«, 

a covariant of the degree 4 in the coefficients, and the degree 12 in the variables; 
and it must vanish when a = 6 = c = 0, e=/=0; this can only be the covariant 

3 (No. 13)»(No. 14) -25 (No. 15)», [= 3^»B - 25C*], 

which it is clear vanishes as required. 

13. In order that the quintic may have four equal roots, or that the system 
may be of the form 41, the simplest type to be considered is 

12.34, 
which gives the function 

2(a-yS)«(7-S)'(^-6y)», 

a covariant of the degree 2 in the coefficients, and of the same degree in the variables; 
this can only be the covariant (No. 14), [=-B]. 

14. Finally, in order that all the roots may be equal, or that the system of 
roots may be of the form 5, the type to be considered is 

12; 
and this gives the function 

a covariant of the degree 2 in the coefficients, and the degree 6 in the variables, 
and this can only be the Hessian (No. 15), [=s C]. 

It will be observed that all the preceding conditions are distinctive; for instance, 
the covariant which vanishes when the system of roots is of the form 311, does not 
vanish when the system is of the form 221, or of any other form not included in 
the form 311. 



151] 



471 



151. 



TABLES OF THE STUEMIAN FUNCTIONS FOR EQUATIONS OF 
THE SECOND, THIRD, FOURTH, AND FIFTH DEGREES. 



[From the Philosophical Transdctiona of the Royal Society of London, vol. CXLVII. for 
the year 1857, pp. 733—736. Received December 18, 1856,— Read January 8, 1857.] 

The general expressions for the Sturmian functions in the form of determinants 
are at once deducible firom the researches of Professor Sylvester in his early papers 
on the subject in the Philosophical Magazine, and in giving these expressions in the 
Memoir 'Nouvelles Recherches sur les Fonctions de M. Sturm,' Liouville, t. xiii. p. 269 
(1848), [65], I was wrong in claiming for them any novelty. The expressions in the 
last-mentioned memoir admit of a modification by which their form is rendered some- 
what more elegant; I propose on the present occasion merely to give this modified 
form of the general expression, and to give the developed expressions of the functions 
in question for equations of the degrees two, three, foUr, and five. 

Consider in general the equation 

Cr = (a, 6, ... j, k^x, 1)», 

and write 

P = (a, 6, ... jJix, ir'\ 

Q=(6, ... j^fcJix, l)--\ 

then supposing as usual that the first coefficient a is positive, and taking for shortness 

th •"" 1 w """ 1 fh "■" 2 
?ij, n,, &c. to represent the binomial coefficients — = — , :.—^ , &c. corresponding 

to the index (n — 1), the Sturmian functions, each with its proper sign, are as 
follows, viz. 



472 



TABLES OF THE STURMIAN FUNCTIOKS FOR 



ri5i 



U. P. 



P. Q 

a , b 



- *P. P. 

'a. • , 
iij6, a , 



*«, Q 
6, . 

fliC^ b i 

I 



+ a^P, xP, P. a»e. xQ. Q . &c 



a. 

I 



a, 
njb. 



a, 
ry:. 



6. 



6. . 
w,c, 6 



where the terms contaimng the powers of x, which exceed the degrees of the several 
functions respectively, vanish identically (as is in fact obvious from the form of the 
expressions), but these terms may of course be omitted ab initio. 

The following are the results which I have obtained; it is well known that the 
last or constant function is in each case equal to the discriminant, and as the 
expressions for the discriminant of equations of the fourth and fifth degrees are given, 
Tables No. 12 and No. 26 [Q', see 143] in my 'Second Memoir upon Quantics'(0. I 
have thought it sufficient to refer to these values without repeating them at length. 

Table for the degree 2. 
The Sturmian functions for the quadric (a, 6, c$a?, 1)* are 




c + 1 \l^x, ly. 



(! a+ I 6 + 1 



5*. 1). 



ac—\ 
6» + l 



Table for the degree 3. 
The Sturmian functions for the cubic (a, 6, c, d$a?, ly are 



a+l 


b + 3 e + 3 (i+1 



5*^ 1)*. 



1 Philotophieal Trantactions, t. cxlvi. p. 101 (1856), [141]. 



151] EQUATIONS OF THE SECOND, THIRD, FOURTH, AND FIFTH DEGREES. 473 



a + 1 


6 + 2 


c+l 



'$.«>, 1)', 




$«. 1). 



1 a'cP 


+ 1 


abed 


+ 6 


' ac" 


-4 


' bd" 


-4 


b'c^ 


-3 



Table for the degree 4. 
The Sturmian functions for the quartic (a, 6, c, d, e$x, 1\ are 



( 



a+ 1 


6 + 4 


c + 6 


(^ + 4 


6 + 1 



5*. 1)*' 



a+ 1 


6 + 3 


c + 3 


c^+l 



$*. 1)'. 



ac— 3 
6^+3 


ac;-3 
6c +3 


ae-\ 
bd-^l 



Ja^. 1)'. 



3( 



a^ce - 1 


a^cfe + 1 


a»c;» + 3 


a6cc —4 


ah^e + 1 


ahd:" -1 


o6c€? — 14 


ac^d +3 


ac» + 9 


6»e +3 


6»c^ + 8 


6»cc; -2 


6V - 6 





$«^ 1), 



aV+1 

Disct. Tab. 
No. 12. 



Table for the degree 5. 
The Sturmian functions for the quintic (a, 6, c, d, e, /$a:, 1)" are 



a + 1 



6 + 5 



c+10 



ef+10 



e + 5 



/+i r^x, i)», 



C. II. 



60 



474 



TABLES OF THE 8TUEM1AN FUNCTIONS, &C. 



[151 



a + 1 


6 + 4 


c + 6 


d-h4 


e+l 



Jix, ly, 



( 



oc - 4 
6« +4 


arf-6 
6c +6 


a« — 4 
bd+i 


a/-l 
6« +1 



"ga:, 1)', 



2( 



a«ce - 8 


a V - 2 


a»c(/* + 3 


a>^ +18 


a«^ +12 


a6c/- 11 


a6»c + a 


a6y+ 2 


abde-- 3 


a6crf- 76 


abce - 42 


ac"e + 8 


ac» +48 


a6rf« -12 


6y + 8 


b^d +40 


cuM +32 


6«cc - 5 


6V -30 


6»e + 30 
b^cd - 20 





\', 1)", 



2( 



aV" - 


2 


a^dp + 


3 


a^def + 


24 


aV/ - 


8 


aV - 


32 


a^bcP - 


11 


a>6y> + 


2 


a^bdef + 


58 


a»Me* + 


264 


a»6c» + 


8 


a'ftcc/" — 


52 


aV«/* + 


104 


a^bd}/-- 


96 


a^ccP/ - 


156 


a^(?df + 


64 


a«ccfe« - 


96 


aVtf> + 


352 


a*cPe + 


108 


a'ccPe - 


938 


a6y* +• 


8 


a»(£* + 


432 


a6*c«/ - 


266 


aJt^ef + 


28 


ab^d^f - 


8 


oftW - 


970 


ah^d^ + 


35 


o^'cPe + 


120 


abi^df -^ 


584 


abi^de + 


2480 


abi?^ + 


120 


aJl^cdf -^ 


264 


a6cc?e - 


360 


ahcd^ - 


1440 


oc*/ - 


288 


a6cy - 


192 


W?dA + 


160 


ac*« - 


960 


6V + 


120 


cu^d^ + 


640 


6»ai/ - 


320 


6*(^ - 


160 


6»ce« - 


75 


6V + 


450 


6»(?6 + 


200 


l^cde - 


1400 


6»cy + 


180 


6»d» + 


800 


6«c»^ - 


100 


6V/ + 


120 






6V6 + 


600 






6Vrf» - 


400 







aV* + l 

+ Ac, 

Disct Tab. 

No. 26, [Q']. 



5^, 1), 



152] 



475 



152, 



A MEMOIR ON THE THEORY OF MATRICES. 



[From the Philosophical Transactions of the Royal Society of London, vol. CXLVili. for 
the year, 1858, pp. 17 — 37. Received December 10, 1857, — Read January 14, 1858.] 

The term matrix might be used in a more general sense, but in the present 
memoir I consider only square and rectangular matrices, and the term matrix used 
without qualification is to be understood as meaning a square matrix ; in this restricted 
sense, a set of quantities arranged in the form of a square, e.g. 

( a , b , c ) 

a' , V , d 

a", V\ d' 

is said to be a matrix. The notion of such a matrix arises naturally firom an 
abbreviated notation for a set of linear equations, viz. the equations 



z = 


ax + 6y '\-cz , 


Y^a'x + Vy + dz , 


Z ^(j^'x-^V'y^c'z, 


may be more simply represented by 


(Z, F, ^ = ( a , 6 , c '^x, y, «), 




a' , V , d 






a", V\ d' 





and the consideration of such a system of equations leads to most of the fundamental 
notions in the theory of matrices. It will be seen that matrices (attending only to 
those of the same order) comport themselves as single quantities; they may be added, 

60—2 



476 



A MEMOIB ON THE THEORY OF MATRICES. 



[152 



multiplied or compounded together, &c. : the law of the addition of matrices is pre- 
cisely similar to that for the addition of ordinar}' algebraical quantities; as regards 
their multiplication (or composition), there is the peculiarity that matrices are not in 
general convertible; it is nevertheless possible to form the powers (positive or negative, 
integral or fractional) of a matrix, and thence to arrive at the notion of a rational 
and integral function, or generally of any algebraical function, of a matrix. I obtain 
the remarkable theorem that any matrix whatever satisfies an algebraical equation of 
its own order, the coeflBcient of the highest power being unity, and those of the 
other powers functions of the terms of the matrix, the last coefiBcient being in bet 
the determinant; the rule for the formation of this equation may be stated in the 
following condensed form, which will be intelligible after a perusal of the memoir, 
viz. the determinant, formed out of the matrix diminished by the matrix considered 
as a single quantity involving the matrix unity, will be equal to zero. The theorem 
shows that every rational and integral function (or indeed every rational function) of 
a matrix may be considered as a rational and integral function, the degree of which 
is at most equal to that of the matrix, less unity; it even shows that in a sense, 
the same is true with respect to any algebraical function whatever of a matrix. One 
of the applications of the theorem is the finding of the general expression of the 
matrices which are convertible with a given matrix. The theory of rectangular 
matrices appears much less important than that of square matrices, and I have not 
entered into it further than by showing how some of the notions applicable to these 
may be extended to rectangular matrices. 

1. For conciseness, the matrices written down at full length will in general be 
of the order 3, but it is to be understood that the definitions, reasonings, and con- 
clusions apply to matrices of any degree whatever. And when two or more matrices 
are spoken of in connexion with each other, it is always implied (unless the contrary 
is expressed) that the matrices are of the same order. 

2. The notation 

( a , 6 , c ^x, y, z) 

a', 6', c' 

represents the set of linear functions 

((a, 6, c\x, y, z\ (a', h\ d\x, y, z\ (a", 6", c"\x, y, z)\ 
so that calling these (X, F, Z), we have 

(X, F, ^ = ( a , 6 , c \x, y, z) 

and, as remarked above, this formula leads to most of the fundamental notions in the 
theory. 



152] 



A MEMOIR ON THE THEORY OF MATRICES. 



477 



3. The quantities {X, Y, Z) will be identically zero, if all the terms of the matrix 

are zero, and we may say that 

( 0, 0, .) 

0, 0, 

0, 0, 
is the matrix zero. 

Again, {X, T, Z) will be identically equal to {x, y, z), if the matrix is 

(1. 0. ) 

0, 1. ! 

0, 0, 1 : 

and this is said to be the matrix unity. We may of course, when for distinctness it 
is required, say, the matrix zero, or (as the case may be) the matrix unity of such an 
order. The matrix zero may for the most part be represented simply by 0, and the 
matrix unity by 1. 

4. The equations 

(Z, F, Z) = { a, b, c Jix.y. z\ X\ F, Z') = ( « , /S , 7 $.^, y, z) 



a' , 6' , c' 

^" J." V 
a , , c 



«'. /3', 7' 
a", /8", 7" 



give 



{X + X, Y+T, Z + Z) = { a +a , b +0 , c +y Jix, y, z) 

a' +a', V +yS', c'+7' 

a" + a", 6" + ^", (Z' + y 
and this leads to 

(o+a, b +P , c+7 ) = (a, b , c ) + (o, fi , 7 ) 



o' +0', b' +ff, c' +7' 
o" + a", 6"+/3", c"+7" 



a' , b' , c' 
a", b", c" 






as a rule for the addition of matrices ; that for their subtraction is of course similar 
to it. 

5. A matrix is not altered by the addition or subtraction of the matrix zero, 
that is, we have M ±0 = M. 

The equation L==M, which expresses that the matrices Z, M are equal, may also 
be written in the form Zcr-M^^O, Le. the difference of two equal matrices is the 
matrix zero. 

6. The equation L^--My written in the form Z + Jf^O, expresses that the sum 
of the matrices L, M \a equal to the matrix zero, the matrices so related are said to be 
opposite to each other ; in other words, a matrix the terms of which are equal but oppo- 
site in sign to the terms of a given matrix, is said to be opposite to the given matrix. 



478 



A MEMOIR ON THE THEORY OF MATRICES. 



[152 



7. It is clear that we have L -{-M = M + L, that is, the operation of addition is 
commutative, and moreover that (L + M) + JV = Z + (M -i-If^^L + M + N, that is, the 
operation of addition is also associative. 



8. The equation 



written under the forms 



(X, Y, Z) — { a , 6 , c $ww?, my, mz) 



(X, T, Z) = m( a , 6 , c $a:, y, z) = ( via , mb , mc ^x, y, z) 



a\ 6' , c' 
a , , c 



ma' , mi/ , mc' 
ma", fii6", m&' 



gives 



m( a y 6, c ) = ( ma , mh , mc ) 



a\ h\ c' 
a'\ h'\ c" 



tna' , mV , mc' 
?7ia", m6", mc" 



as the rule for the multiplication of a matrix by a single quantity. The multiplier m 
may be written either before or after the matrix, and the operation is therefore com- 
mutative. We have it is clear m{L-\-M) — mL + mJf, or the operation is distributive. 

9. The matrices L and mL may be said to be similar to each other; in 
particular, if m = 1, they are equal, and if m = — 1, they are opposite. 

10. We have, in particular, 

m ( 1, 0, ) = ( m, 0, ), 



0, m, 
0, 0, m 



0, 1, 

0, 0, 1 

or replacing the matrix on the left-hand side by unity, we may write 

m=={m, 0, ); 

0, m, 

0, 0, m 

the matrix on the right-hand side is said to be the single quantity m considered as 
involving the matrix unity. 



11. The equations 
(Z, F, Z) = ( a, 



a , 






6 , c $a?, y, z\ (a?, y, ^) = ( a , ^ , 7 $f, 1;, {), 
6', C a', /3', y 

i", c" a", /3", y 



152] 



A MEMOIR ON THE THEORY OF MATRICES. 



479 



give 



(X,7.Z) = (A, B, C IJf , 17. ?) = ( o , 6, c ^a. 



A', B, C 
A", B", C" 



a'. V, 
a , 6 , 



jf 



a, 






fjt Q>l ^11 



and thence, substituting for the matrix 

{A , B , C ) 

A', R. C 

I A", F', C" 
its value, we obtain 

{{a.b.c^a. a', a"), {a,b ,c^P, &, /8"). (a , 6 . c 57, 7. 7") ) - ( « . & . c $ a . ^ . 7 ) 



(a' .b'.c' $a, a', O, (a' . 6' , c' 3[y9, ff, /3"), (a' , 6' . c' $7, 7'. 7") 
(a", 6", c"$a, a', a"), (a". 6", c"$/8, ff, n. (o", 6". c"$7. 7'. 7") 



o',6',c' 

_// 1;// -^ 



«',/3'.7' 
a", /8". 7" 



as the rule for the multiplication or composition of two matrices. It is to be 
observed, that the operation is not a commutative one; the component matrices may 
be distinguished as the first or further component matrix, and the second or nearer 
component matrix, and the rule of composition is as follows, viz. any line of the com- 
pound matrix is obtained by combining the corresponding line of the first or further 
component matrix successively with the several columns of the second or nearer com- 
pound matrix. 

[We may conveniently write 

(a. «', a"). (A /9'. ^'). (7. 7'. 7") 



(a , 


b. 


c) 


if 


f> 


ij 


(a', 


b', 


c') 


t> 


it 


if 


(0". 


b". 


c") 


n 


» 


n 



to denote the left-hand side of the last preceding equation.] 

12. A matrix compounded, either as first or second component matrix, with the 
matrix zero, gives the matrix zero. The case where any of the terms of the given 
matrix are infinite is of course excluded. 

13. A matrix is not altered by its composition, either as first or second component 
matrix, with the matrix unity. It is compounded either as first or second component 
matrix, with the single quantity m considered as involving the matrix unity, by 
multiplication of all its terms by the quantity m:. this is in fact the before-mentioned 
rule for the multiplication of a matrix by a single quantity, which rule is thus seen 
to be a particular case of that for the multiplication of two matrices. 

14. We may in like manner multiply or compound together three or more 
matrices: the order of arrangement of the factors is of course material, and we may 



480 



A MEMOIR ON THE THEORY OF MATRICES. 



[152 



distinguish them as the first or furthest, second, third, &c., and last or nearest 
component matrices: any two consecutive factors may be compounded together and 
replaced by a single matrix, and so on until all the matrices are compound^ together, 
the result being independent of the particular mode in which the composition is 
effected; that is, we have L.MN^LM .N ^ LMN, LM.NP^L. MN.P, &a, or the 
operation of multiplication, although, as already remarked, not commutative, is associative. 

15. We thus arrive at the notion of a positive and integer power 2> of a 
matrix L, and it is to be observed that the different powers of the same matrix are con- 
vertible. It is clear also that p and q being positive integers, we have L^.L^^L^^^, 
which is the theorem of indices for positive integer powers of a matrix. 

16. The last-mentioned equation, 2>.Z« = 2>+«, assumed to be true for all values 
whatever of the indices p and q, leads to the notion of the powers of a matrix for any 
form whatever of the index. In particular, 2> . Z® = J!> or Z<* = 1, that is, the 0th power 
of a matrix is the matrix unity. And then putting p = l, } = — 1, or /> = — !, g=l, we 
have L . L"^ = Z~^ . Z = 1 ; that is, L~\ or as it may be termed the inverse or reciprocal 
matrix, is a matrix which, compounded either as first or second component matrix 
with the original matrix, gives the matrix unity. 

17. We may arrive at the notion of the inverse or reciprocal matrix, directly 
from the equation 

(Z, F. Z) = ( a , 6 , c Jxy y, z\ 



a' , 6' , c' 

«" Uf V 

a , by c 



in fact this equation gives 



{XyyyZ) = (A, A\ ^"$Z, F, Z) = (( a, 6, c )-^$Jr, F, Z), 



5, B, B' 



a , , c 



and we have, for the determination of the coefficients of the inverse 
matrix, the equations 

(A. A', ^"$o, 6, c ) = ( 1. 0, ), 



or reciprocal 



B, ff, B' 

C, C, C" 



a', V, c' 
a", b", c" 



0. 1, 
0, 0, 1 



(a . b, c 11 A, A', A" ) = ( 1, 0, ), 



a' , 6' , c' 



a , 



6", 



.// 



B, B, B' 

C, C, C" 



0, 1, 
0, 0. 1 



s 



152] 



A MEMOIR ON THE THEORY OF MATRICES. 



481 



which are equivalent to each other, and either of them is by itself sufficient for the 
complete determination of the inverse or reciprocal matrix. It is well known that if 
V denote the determinant, that is, if 

V = a , 6 , c 

a' , 6' , c' 

^" J." «" 
a , , c 

then the terms of the inverse or reciprocal matrix are given by the equations 



V 



1, 0, 
0, b', c' 
0, h". c" 



B = l 
V 



, 1, 
a' , 0, c' 
a", 0, c" 



, &c. 



or what is the same thing, the inverse or reciprocal matrix is given by the equation 

( a , 6 , c )-^ 1 ( BaV, Ba'V, a„^ V ) 



a', b\ c' 



SfrV, a,,.v, a^-v 
ScV, ac'V, a,..v 



where of course the differentiations must in every case be performed as if the terms 
a, 6, &c. were all of them independent arbitrary quantities. 

18. The formula shows, what is indeed clear d priori, that the notion of the 
inverse or reciprocal matrix fails altogether when the determinant vanishes: the matrix 
is in this case said to be indeterminate, and it must be understood that in the 
absence of express mention, the particular case in question is frequently excluded from 
consideration. It may be added that the matrix zero is indeterminate; and that the 
product of two matrices may be zero, without either of the factors being zero, if only 
the matrices are one or both of them indeterminate. 

19. The notion of the inverse or reciprocal matrix once established, the other 
negative integer powers of the original matrix are positive integer powers of the 
inverse or reciprocal matrix, and the theory of such negative integer powers may be 
taken to be known. The theory of the fractional powers of a matrix will be ftirther 
discussed in the sequel. 

20. The positive integer power L^ of the matrix L may of course be multiplied 
by any matrix of the same degree: such multiplier, however, is not in general con- 
vertible with L; and to preserve as far as possible the analogy with ordinary 
algebraical frmctions, we may restrict the attention to the case where the multiplier 
is a single quantity, and such convertibility consequently exista We have in this 
manner a matrix cL^, and by the addition of any number of such terms we obtain 
a rational and integral function of the matrix L. 

a 11. 61 



482 A MEMOIR ON THE THEORY OF MATRICES. [132 

21. The general theorem before referred to will be best anderstood by a com- 
plete deTelopment of a particular case. Imagine a matrix 

JI={ a, 6 ), 

c, d 
and form the determinant 

a — M, h 

c ,d-M 
the developed expression of this determinant is 

Jf » - ( a + rf) JT + (ad - 6c) Jf • ; 
the values of M\ M\ Jf* are 

( a^^hc , 6(a + rf) ), ( a. 6 ). ( 1. X 

I ' i 

cia-^-d), (f + 6c ' c, d\ ! 0, 1 i 

and substituting these values the determinant becomes equal to the matrix zero, vis. 
we have 

a-Jf, 6 =(a» + ftc, 6(a + ci) )-(a + rf) ( a, 6 ) + (ad-ftc) ( 1, 0) 



c ,d-M\ I c(a + d), cf + ftc ! c, d ^ 0, 1 I 

= ( (a» + ftc)-(a + ci)a + (a(i-6c), 6(a + d)-(a + rf)6 ) = ( 0, ); 

c(a + d)-(a + d)c , cf + ftc-(a + cOd+arf-6c ! ! 0, I 

that is 

a - Jf , 6 . = 0, 

c ,d^M 
where the matrix of the determinant is 

( a, 6 )-i/( 1, ), 

c, d I I 0, 1 I 

that is, it is the original matrix, diminished by the same matrix considered as a single 
quantity involving the matrix unity. And this is the general theorem, viz. the deter- 
minant, having for its matrix a given matrix less the same matrix considered as a 
single quantity involving the matrix unity, is equal to zero. 

22. The following symbolical representation of the theorem is, I think, worth 

noticing: let the matrix if, considered as a single quantity, be represented by JB, then 

writing 1 to denote the matrix unity, S,l will represent the matrix M, considered 

as a single quantity involving the matrix unity. Upon the like principles of notation, 

I. if will represent, or may be considered as representing, simply the matrix if, and 

the theorem is 

Det (I.if-iBf.l) = 0. 



152] 



A MEMOIR ON THE THEORY OF MATRICES. 



483 



23. I have verified the theorem, in the next simplest case of a matrix of the 
order 3, viz. if Jf be such a matrix, suppose 

if = ( a, 6, c ), 
d, e, f 



then the derived determinant vanishes, or we have 



a — M, b 
d , e 
9 , h 



> c 
M, f 
, i-if 



= 0, 



or expanding 
Jf* — (a + e + i) if ^ + (^' + ta + 06 —/A — eg — bd) M — {ad + bfg + cdh — afh — bdi — ceg) = ; 

but I have not thought it necessary to undertake the labour of a formal proof of 
the theorem in the general case of a matrix of any degree. 

24. If we attend only to the general form of the result, we see that any matrix 
whatever satisfies an algebraical equation of its own order, which is in many cases the 
material part of the theorem. 

25. It follows at once that every rational and integral function, or indeed every 
rational function of a matrix, can be expressed as a rational and integral function of 
an order at most equal to that of the matrix, less unity. But it is important to 
consider how far or in what sense the like theorem is true with respect to irrational 
functions of a matrix. If we had only the equation satisfied by the matrix itself, 
such extension could not be made; but we have besides the equation of the same 
order satisfied by the irrational function of the matrix, and by means of these two 
equations, and* the equation by which the irrational function of the matrix is deter- 
mined, we may express the irrational function as a rational and integral function of 
the matrix, of an order equal at most to that of the matrix, less unity; such expression 
will however involve the coefficient of the equation satisfied by the irrational function, 
which are functions (in number equal to the order of the matrix) of the terms, 
assumed to be unknown, of the irrational function itself. The transformation is never- 
theless an important one, as reducing the number of unknown quantities firom n' (if n 
be the order of the matrix) down to n. To complete the solution, it is necessary to 
compare the value obtained as above, with the assumed value of the irrational function, 
which will lead to equations for the determination of the n unknown quantitiea 

26. As an illustration, consider the given matrix 



M=( a, 6 ), 
c, d 



61—2 



484 



A MEMOIR ON THE THEORY OF MATRICES. 



[152 



and let it be required to find the matrix L = *JM, In this case M satisfies the equation 



and in like manner if 



then L satisfies the equation 



if»-(a + d)Jf + ad-6c = 0; 

7. S 
Z« - (a + S) Z + aS- /87 = ; 



and from these two equations, and the rationalized equation Z' = M, it should be possible 
to express L in the form of a linear function of M: in fact, putting in the last 
equation for Z" its value (=M), we find at once 



Z = 



i-g[.lf + (aS-^7)], 



which is the required expression, involving as it should do the coefficients a + S, aS^/3y 
of the equation in L. There is no difficulty in completing the solution; write for 
shortness a + S = X, aS — ySy = F, then we have 



Z = ( 0, fi ) = ( a'\-Y 

X 



b_ ), 
X 

d+T 



and consequently forming the values of a + S and aS — ^87, 



X = 



a+d+2Y 



y_ (a+r)(d+r)-6c 
and putting also a + d = P, ad — bc^Q, we find without difficulty 

and the values of a, /8, 7, S are consequently known. The sign of ^Q is the same in 
both formulae, and there are consequently in all four solutions, that is, the radical vif 
has four values. 

27. To illustrate this further, suppose that instead of M we have the matrix 

if> = ( a, 6 )» = ( d' + bc , 6(a + d) ), 



c, d 



c(a + d), d» + 6c 



152] 



A MEMOIR ON THE THEORY OF MATRICES. 



485 



so that D = M^, we find 



P = (aH-d)«-2(ad-6c), 



and thence VQ = ± (ad — be). Taking the positive sign, we have 



and these values give simply 



But taking the negative sign, 



F= ad — be, 
X=±(a + d), 

L=±^( a, b )=±M, 
I c, d 

Y=-ad + bCy 

X = ± V(a - d)« + 46c, 



and retaining X to denote this radical, we find 



Z = 



_( a'-ad+26c b(a + d) ), 



X 

c(a'{-d) 



X 
d^-ad + 2bc 



which may also be written 



, a+d(a, b) 2(ad-6c) ( 1, ), 

^ I 0, 1 I 



or, what is the same thing. 



c, d 



J ^a + d ^ 2 (ad — be) 
" Z X ' 



and it is easy to verify d posteriori that this value in fact gives D = M\ It may 
be remarked that if 

if» = ( 1, )» = 1. 

0, 1 

the last-mentioned formula fails, for we have X = ; it will be seen presently that 
the equation Z* = 1 admits of other solutions besides Z = ± 1. The example shows how 
the values of the fractional powers of a matrix are to be investigated. 

28. There is an apparent difficulty connected with the equation satisfied by a 
matrix, which it is proper to explain. Suppose, as before, 

M^( a, 6 ), 
c, d 



486 



A MEMOIB ON THE THEORY OF MATRICES. 



[152 



so that M satisfies the equation 

a - If, 6 =0, 

and let X^, X„ be the single quantities, roots of the equation 



or 



c d — X 



= 



or 



Z«-(a+(i)Z + ad-6c=0. 



The equation satisfied by the matrix may be written 

(if-Z,)(if-Z„) = 0. 

in which X,, X,, are to be considered as respectively involving the matrix unity, and it 
would at first sight seem that we ought to have one of the simple fiEu^tors equal to 
zero; this is obviously not the case, for such equation would signify that the perfectly 
indeterminate matrix M was equal to a single quantity, considered as involving the 
matrix unity. The explanation is that each of the simple factors is an indeterminate 
matrix, in fact M — X, stands for the matrix 



(a-X„ 



b 
d-X 



). 



and the determinant of this matrix is equal to zero. The product of the two factors 
is thus equal to zero without either of the factors being equal to zero. 

29. A matrix satisfies, we have seen, an equation of its own order, involving the 
coefficients of the matrix; assume that the matrix is to be determined to satisfy some 
other equation, the coefficients of which are given single quantities. It would at first 
sight appear that we might eliminate the matrix between the two equations, and thus 
obtain an equation which would be the only condition to be satisfied by the terms 
of the matrix ; this is obviously wrong, for more conditions must be requisite, and we 
see that if we were then to proceed to complete the solution by finding the value of 
the matrix common to the two equations, we should find the matrix equal in every case 
to a single quantity considered as involving the matrix unity, which it is clear ought 
not to be the case. The explanation is similar to that of the difficulty before adverted 
to; the equations may contain one, and only one, common factor, and may be both of 
them satisfied, and yet the common factor may not vanish. The necessary condition 
seems to be, that the one equation should be a factor of the other; in the case where 
the assumed equation is of an order equal or superior to the matrix, then if this 
equation contain as a factor the equation which is always satisfied by the matrix, the 
assumed equation will be satisfied identically, and the condition is sufficient as well 
as necessary: in the other case, where the assumed equation is of an order inferior 
to that of the matrix, the condition is necessary, but it is not sufficient. 



152] 



A MEMOIR ON THE THEORY OF MATRICES. 



487 



30. The equation satisfied by the matrix may be of the form M^ = 1 ; the 
matrix is in this ease said to be periodic of the ?ith order. The preceding conside- 
rations apply to the theory of periodic matrices; thus, for instance, suppose it is 
required to find a matrix of the order 2, which is periodic of the second order. Writing 

Jf=( a, 6 ), 
c, d 



we have 



and the assumed equation is 



Jlf»-(a + d)if+a(i-6c = 0, 



:ilf « - 1 = 0. 



These equations will be identical if 

a + d = 0, a(i - 6c = — 1, 

that is, these conditions being satisfied, the equation Jf * — 1 = required to be satisfied, 
will be identical with the equation which is always satisfied, and will therefore itself 
be satisfied. And in like manner the matrix M of the order 2 will satisfy the 
condition Jf'— 1 = 0, or will be periodic of the third order, if only Jf'— 1 contains as 
a factor 

if " - (a + d) -M'+ a(i - 6c, 
and so on. 

31. But suppose it is required to find a matrix of the order 3, 



M^{ a, 
d, 



6, c) 
e. f 



which shall be periodic of the second order. Writing for shortness 



a — Jf, 



h 



, 

M, f 



^-iM^-AM^ + BM^C), 



the matrix here satisfies 



M*-AM* + BM-C = 0, 



and, as before, the assumed equation is Jf ' — 1 = 0. Here, if we have 1+J5 = 0, A + C=^0, 
the left-hand side will contain the £Ekctor (if'— 1), and the equation will take the form 
(ilf* — 1) (if + C) = 0, and we should have then if' —1 = 0, provided M+C were not an 
indeterminate matrix. But M+C denotes the matrix 



( 



a+C, b 



) 



d 
9 



e + C. f 

h , i + C 



488 



A MEMOIR ON THE THEORY OF MATRICES. 



[152 



the determinant of which is C* + -4C*+J5(7+ C, which is equal to zero in virtue of 
the equations 1+-B = 0, -44-C = 0, and we cannot, therefore, from the equation 
(if»-l)(if+C) = 0, deduce the equation if«-l = 0. This is as it should be, for the 
two conditions are not sufficient, in fact the equation 



Jf« = 



( a^ +bd +cgy ab + be + ch, ac + bf+ d ) 

da + ed +fg, db + ei^ -^fh, dc + ef-¥fi 

ga +hd + ig, gb + he + ih, gc + hf+ i* 



= 1 



gives nine equations, which are however satisfied by the following values, involving in 
reality four arbitrary coefficients; viz. the value of the matrix is 



( 



- (7 -f g) fiv-' 

a + /8 + 7 

a + /8 + 7 



^(0 + y)vfl'" ^(0+y)vfjr^ ) 

a + /8 + 7 

- (7 + g) Xfi-^ 
a + fi + y 



o + i8 + 7 


/9 


a + 13 + y 


- (0 + /3) I/X-' 



04-/8 + 7 



a + /8 + 7 



so that there are in all five relations (and not only two) between the coefficients of 
the matrix. 

32. Instead of the equation M^ — 1=0, which belongs to a periodic matrix, it is 
in many cases more convenient, and it is much the same thing to consider an 
equation M^ — k = 0, where A; is a single quantity; The matrix may in this case be 
said to be periodic to a factor pris. 

33. Two matrices L, M are convertible when LM = ML. If the matrix M is given, 
this equality affords a set of linear equations between the coefficients of L equal in 
number to these coefficients, but these equations cannot be all independent, for it is 
clear that if X be any rational and integral function of M (the coefficients being single 
quantities), then L will be convertible with Jf ; or what is apparently (but only appa- 
rently) more general, if L be any algebraical function whatever of M (the coefficients 
being always single quantities), then L will be convertible with M. But whatever the 
form of the function is, it may be reduced to a rational and integral function of an 
order equal to that of M, less unity, and we have thus the general expression for 
the matrices convertible with a given matrix, viz. any such matrix is a rational and 
integral function (the coefficients being single quantities) of the given matrix, the 
order being that of the given matrix, less unity. In particular, the general form of 
the matrix L convertible with a given matrix M of the order 2, is Z = aAf + /8, or 
what is the same thing, the matrices 

( a, 6 ), ( a\ V ) 



c, d 



c', d' 



will be convertible if a' — d' : 6' : c' = a — d : 6 : c. 



152] 



A MEMOIR ON THE THEORY OF MATRICES, 



489 



34. Two matrices Z, M are skew convertible when LM = — ML ; this is a relation 
much less important than ordinary convertibility, for it is to be noticed that we cannot 
in general find a matrix L skew convertible with a given matrix M. In fact, con- 
sidering M as given, the equality affords a set of linear equations between the coeffi- 
cients of L equal in number to these coefficients; and in this case the equations are 
independent, and we may eliminate all the coefficients of Z, and we thus arrive at a 
relation which must be satisfied by the coefficients of the given matrix M, Thus, 

suppose the matrices 

( a, 5 ), ( a\ V ) 



c, 



c\ d! 



are skew convertible, we have 



(a, h){a\ 6' ) = ( oa' + fcc', oJ' + W ), 



0, d \ d y dl 



ca' + dc', cV + dd! 



(a\ 5' )( a, 5) = (oa' + 6'c, a'5+6'rf), 



c\ d! \ c, d 



and the conditions of skew convertibility are 



c'a + d'c, c'b + d'd 



2oa' + 5c' + b'c = 0, 

5' (a +rf)+6(a'+d') = 0, 

c'{a +d) +c(a' + dO = 0» 
2dd' + bc' + b'c =0. 

Eliminating a\ b\ c\ d\ the relation between a, 5, c, d is 



2a. 


c , 


b , . 


b , 


a +d, 


b 


c , 


• 


a + d, c 


• 


c , 


b , 2d 



= 0, 



which is 



(a + dy (ad-bc)-^ 0. 



Excluding from consideration the case orf — 6c = 0, which would imply that the matrix 
was indeterminate, we have a + d = 0. The resulting system of conditions then is 

a + d = 0, a' + d' = 0, aa' + 6c' + 5'c + dd' = 0, 

the first two of which imply that the matrices are respectively periodic of the second 
order to a factor pris, 

35. It may be noticed that if the compound matrices LM and ML are similar, 
they are either equal or else opposite ; that is, the matrices L, M are either convertible 
or skew, convertible. 

C. II. 62 



490 



A MEMOIR ON THE THEORY OF MATRICES. 



[152 



36. Two matrices such as 



( a, 6 ), ( a, c ), 



c, d 



6, d 



are said to be formed one fix)m the other by transposition, and this may be denoted 
by the symbol tr. ; thus we may write 

(a, c ) = tr. ( a, 6 ). 



6, d \ 



c, d 



The effect of two successive transpositions is of course to reproduce the original matrix. 



37. It is easy to see that if M be any matrix, then 

(tr. My = tr. (MP), 

(tr. M)-' = tr. ( Jf-0. 



and in particular, 



38. If Z, if be any two matrices, 

tr. (Zi/) = tr. il/. tr.Z, 
and similarly for three or more matrices, L, M, N, &c., 

tr. (LMN) == tr. N. tr. M, tr. i, &c. 

40. A matrix such as 

(a, A, g ) ' 

K 6, / 

which is not altered by transposition, is said to be symmetrical. 

41. A matrix such as 

( 0, V, -m) 

-V, 0, X I 
A^ -X, ! 

which by transposition is changed into its opposite, is said to be skew symmetrical. 

42. It is easy to see that any matrix whatever may be expressed as the sum of 
a s}rmmetrical matrix, and a skew sjrmmetrical matrix ; thus the* form 

h-v, 6., / + X 
flr + M, /-X, c 

which may obviously represent any matrix whatever of the order 3, is the sum of the 
two matrices last before mentioned. 



152] 



A MEMOIB ON THE THEORY OF MATRICES. 



491 



43. The following formulae, although little more than examples of the composition 
transposed matrices, may be noticed, viz. 

(a, 6 $ a, c ) = ( a' + 6', ac-{-bd ) 



c, d 



d, 5 



ac-^bd, c' + d' 



which shows that a matrix compounded with the transposed matrix gives rise to a 
symmetrical matrix. It does not however follow, nor is it the fact, that the matrix and 
transposed matrix are convertible. And also 

(a, c ^ a, 5 $ a, c ) = ( a» + 6cd + a(6» + c'), c» +a6d + c(a« + cP) ) 



b, d c, d \ \ b, d 

which is a remarkably symmetrical form. It is needless to proceed further, since it 
is clear that 

(a, c ^ a, 6 $ «> ^ a, 6 )=(( a, c 1^ a, b ))*. 



b, d \ c, d b, d c, d 



b, d Cf d 



44. In all that precedes, the matrix of the order 2 has frequently been con- 
sidered, but chiefly by way of illustration of the general theory ; but it is worth while to 
develope more particularly the theory of such matrix. I call to mind the fundamental 
properties which have been obtained, viz. it was shown that the matrix 



satisfies the equation 

and that the two matrices 



M = ( a. 5 ), 

1 c, a ! 

M*-{a + d)M + ad-bc = 0, 
(a, b). ( a\ V ), 



c, d 



will be convertible if 



c', d' 



a! — d' : V ', c' ^a^d : b : Ct 



and that they will be skew convertible if 

the first two of these equations being the conditions in order that the two matrices 
may be respectively periodic of the second order to a factor jyrhs, 

45. It may be noticed in passing, that if Z, Jlf are skew convertible matrices of 
the order 2, and if these matrices are also such that i> = — 1, ilf' = — 1, then putting 
JV=Z3f= — JfL, we obtain 

Z«=-l. if«=-l, i<r« = -i, 

L^MN^^NM, M^^NL^^NL, N^LM^-ML, 

which is a system of relations precisely similar to that in the theory of quatemiona 

62—2 



492 



A MEMOIR ON THE THEORY OF MATRICES. 



[152 



46. The integer powers of the matrix 

Jf = ( a, 5 \ 
c, d 

are obtained with great facility from the quadratic equation ; thus we have, attending 
first to the positive powers, 

Jtfa = (a 4- d)M-(ad - be), 

iV» = [(a + d)« - (orf - 6c)] if - (a + d) (orf - 6c), 

&c., 

whence also the conditions in order that the matrix may be to a factor pris periodic of 
the orders 2, 3, &c. are 

a + d * = 0, 

(a + dy - (ad - 6c) = 0, 

&c. ; 

and for the negative powers we have 

(od- 6c) -¥-' = - il/ + (a + d), 

which is equivalent to the ordinary form 

{ad''bc)M-' = { . d, -6 ); 

-c, a I 

and the other negative powers of M can then be obtained by successive multiplications 
with M-\ 

47. The expression for the nth power is however most readily obtained by means 
of a particular algorithm for matrices of the order 2. 

Let A, 6, c, J, q he any quantities, and write for shortness iJ = — A' — 46c; suppose 
also that h\ h\ c', tT, q' are any other quantities, such nevertheless that A' : 6' : c' = A : 6 : c, 

and write in like manner iJ* = — A'* — 46V. Then observing that —r^ . -—r . — are 
respectively equal to -f=^,> "7=57 » "7^* *^^ matrix 



{ j( , h\ 2hJ 



) 



2cJ 
'JR 



.J (cot, + -4) 



contains only the quantities J, q, which are not the same in both systems; and we 
may therefore represent this matrix by (J, q\ and the corresponding matrix with 



152] 



A MEMOIR ON THE THEORY OF MATRICES. 



493 



h\ h\ c\ J\ q' by {J\ q'). The two matrices are at once seen to be convertible (the 
assumed relations h' : h' : c' — h : h : c correspond in fact to the conditions, 

a'^d! : b* : c' = a — d : b : c, 
of convertibility for the ordinary form), and the compound matrix is found to be 



\sin or sm a ^ ^ / 



and in like manner the several convertible matrices {J, q), (J\ q'\ (J'\ q") &c. give 
the compound matrix 



Vsinasma smg ... ^ ^ i j 



48. The convertible matrices may be given in the first instance in the ordinary^ 
form, or we may take these matrices to be 

( a, 6 ), ( a\ V ), ( a", 6" ), &c. 



c, d 



c\ d' 



c\ d" 



where of course d — a : b : c = d* — a' : b' : cf = d" — a" : b'* : c** ^ &c. Here writing 
h = d — a, and consequently R = — {d — o,y — 46c, and assuming also «/ = J VS and 

cot a = — , — , and in like manner for the accented letters, the several matrices are 



respectively 



(i -^R. q) (4 >^, q'), (i ViP'. q"), &c.. 



and the compound matrix is 



/ 8in( «y + q' + q" ^ ^^ ^g> ^ ^ + ,' + y"+ ...) . 

49. When the several matrices are each of them equal to 

(a, 5 ), 
c, d 

we have of course g = g' = g" . . . , R = R = Itf\,, , and we find 



( a, 6 )- 
c, c2 



=(s«^. "5)^ 



or substituting for the right-hand side, the matrix represented by this notation, and 
putting for greater simplicity 



^p (i v:r)» = (J m L,oT L= ^l^i"? (i vs)»-. 



494 A MEMOIR ON THE THEORY OF BfATRICE& [152 

we find 

( a. b )»=:(iZ(v^cotn9-(d-a)), U ) 

\ c, d \ Lc , i Z (Vii cot wj + (d — a)) ; 

where it will be remembered that 

-B = — (d — a)* — 46c and cot 9 = ._ , 

the last of which equations may be replaced by 

. J — 1 . d + a + V^TS 
cosg + v — l8ma= 7- , 

^ ^ 2Vad-6c 

The formula in £Ekct extends to negative or fractional values of the index n, and when 
n is a fraction, we must, as usual, in order to exhibit the formula in its proper 
generality, write q + 2nnr instead of q. In the particular case n » ^, it would be easjr 
to show the identity of the value of the square root of the matrix with that before 
obtained by a different process. 

50. The matrix will be to a factor pris, periodic of the nth order if only sinitf^O, 
that is, if 9 = - (m must be prime to n, for if it were not, the order of periodicity 

would be not n itself, but a submultiple of n) ; but cos q = — . , and the condition 

2 V od — be 

is therefore 

(d + o)»- 4 (ad -6c) COS' "^ = 0, 

or as this may also be written, 

(P-^a^^ 2ad cos h 46c cos* — = 0, 

n n 

a result which agrees with those before obtained for the particular values 2 and 3 
of the index of periodicity. 

51. I may remark that the last preceding investigations are intimately connected 

QX -\~ h 
with the investigations of Babbage and others in relation to the function <f)X = -^ . 

I conclude with some remarks upon rectangular matrices. 

52. A matrix such as 

( a, 6, c ) 

I a', 6', c I 

where the number of columns exceeds the number of lines, is said to be a broad 

matrix ; a matrix such as 

( a, 6 ) 

where the number of lines exceeds the number of columns, is said to be a deep matrix. 



152j 



A MEMOIR ON THE THEORY OF MATRICES. 



495 



53. The matrix zero subsists in the present theory, but not the matrix unity. 
Matrices may be added or subtracted when the number of the lines and the number 
of the columns of the one matrix are respectively equal to the number of the lines 
and the number of the columns of the other matrix, and under the like condition 
any number of matrices may be added together. Two matrices may be equal or 
opposite, the one to the other. A matrix may be multiplied by a single quantity, 
giving rise to a matrix of the same form ; two matrices so related are similar to 
each other. 

54. The notion of composition applies to rectangular matrices, but it is necessary 
that the number of lines in the second or nearer component matrix should be equal 
to the number of columns in the first or further component matrix; the compound 
matrix will then have as many lines as the first or further component matrix, and 
as many columns as the second or nearer component matrix. 

55. As examples of the composition of rectangular matrices, we have 
{a.b,cji a\ b\ c\ d' ) = ( (a, 5, cJia\ e\ iO> («. b, c$h\f\ f) (a, 6, cl^c^ g\ k'\ (a, 6, c$d', K V) ), 



d,ej 



e\f\g\K 



(d, e,/$< e\ i% (d, e,/$6', /, /) (d, e,/$c', fir', ^ (d, 6,/K. K V) 



and 



( a, d $ a', V, d, d') ) = ( (a. d$a'. e'). («. ^F. A («. ^<^> 9'\ («. ^d', h') ). 

(6, e$a'. e'), (h, eji'./), (6. e$c'. (/'), {h. ejd', h!) 
(c, /$a'. e'), (c. /$6', /), (c, /$c'. 9')> (c /K. A') 



c./l 



56. In the particular case where the lines and columns of the one component 
matrix are respectively equal in number to the columns and lines of the other com- 
ponent matrix, the compound matrix is square, thus we have 

( o, 6, c $ a', d' ) = ( (a. h, c$o', h', c'), (a, h, c^d', e', f) ) 



d, e, / 



V, e' 



(d. e. f^a', b', cO, (d, e, f^d', e', /') 



and 



( a', d' $ a, 6, c ) = ( (a', d'$a, d), (a', d'$6, e), (a', d'^c, f) ). 



V, e' 
c'.f 



d, e,f 



{V, ii \a, d). (6', e' $6, e), (6', c' $c, /) 
(c', f\a, d), (c', f'\h, e). (c'. /'$c, /) 



The two matrices in the case last considered admit of composition in the two different 
orders of arrangement, but as the resulting square matrices are not of the same order, 
the notion of the convertibility of two matrices does not apply even to the case in 
question. 

57. Since a rectangular matrix cannot be compounded with itself, the notions of 
the inverse or reciprocal matrix and of the powers of the matrix and the whole resulting 
theory of the functions of a matrix, do not apply to rectangular matrices. 



496 



A MEMOIR ON THE THEORY OF MATRICES. 



[152 



58. The notion of transposition and the symbol tr. apply to rectangular matrices, 
the effect of a transposition being to convert a broad matrix into a deep one and 
reciprocally. It may be noticed that the symbol tr. may be used for the purpose of 
expressing the law of composition of square or rectangular matrices. Thus treating 
(a, 6, c) as a rectangular matrix, or representing it by (a, 6, c), we have 

tr. ( a\ b\ c' ) = ( a ), 



and thence 



( a, b, c ) tr. ( a\ h\ c' ) = (a, 6, cY a' ) 



= (a, 6, c\a\ h\ c'), 



so that the symbol 



(a, 6, c\a\ b\ c*) 



would upon principle be replaced by 

( a, 5, c ) tr. (a', 5', c' ): 

I III 

it is however more convenient to retain the symbol 

(a, 6, c^a\ b\ c^). 
Hence introducing' the symbol tr. only on the left-hand sides, we have 

^ a, 6, c ) tr. ( a, b\ cM = ( (a, 6, cja\ b\ c'), (a, 6. c$d', e\ f) ), 
d, ^. / I i d\ ^, /' I I (d. e, fla\ b\ c% (d, e, fJid\ ^. /O | 



or to take an example involving square matrices. 



( a, 5 ) tr. ( a', 6' ) = ( (a, 6$a', 5'), (a, 6$d', e') ) ; 

I d, e I I d', e' I I (d, c$a', 6'), (d, e^d\ e') 



it thus appears that in the composition of matrices (square or rectangular), when the 
second or nearer component matrix is expressed as a matrix preceded by the symbol 
tr., any line of the compound matrix is obtained by compounding the corresponding 
line of the first or further component matrix successively with the several lines of the 
matrix which preceded by tr. gives the second or nearer component matrix. It is clear 
that the terms 'symmetrical* and *skew symmetrical* do not apply to rectangular 
matrices. 



153] 



497 



153. 



A MEMOIR ON THE AUTOMORPHIC LINEAR TRANSFORMATION 

OF A BIPARTITE QUADRIC FUNCTION. 



[From the Philosophical Transactions of the Royal Society of London, vol. cxLvm. for the 
year 1858, pp. 39 — 4!6. Received December 10, 1857, — Read January 14, 1858.] 

The question of the automorphic linear transformation of the function a^ •\- y^ •¥ s^, 
that is the transformation by linear substitutions, of this function into a function 
a?/ + y/ + V of the same form, is in eflfect solved by some formulae of Euler's for the 
transformation of coordinates, and it was by these formulae that I was led to the 
solution in the case of the sum of n squares, given in my paper "Sur quelques pro- 
pri^tds des determinants gauches"(0- ^ solution grounded upon an d priori investiga- 
tion and for the case of any quadric function of n variables, was first obtained by 
M. Hermite in the memoir "Remarques sur une M^moire de M. Cayley relatif aux 
determinants gauches"(')- This solution is in my Memoir "Sur la transformation d'une 
function quadratique en elle-meme par des substitutions lin6aires"('), presented under a 
somewhat different form involving the notation of matrices. I have since found that 
there is a like transformation of a bipartite quadric function, that is a lineo-linear 
function of two distinct sets, each of the same number of variables, and the develop- 
ment of the transformation is the subject of the present memoir. 

1. For convenience, the number of variables is in the analytical formulae taken 
to be 3, but it will be at once obvious that the formulae apply to any number of 
variables whatever. Consider the bipartite quadric 

( a , 5 , c $a?, y, ^][x, y, z), 

a' , 6' , c' 

a y , c 

1 CreUe, X. xxxn. (1846) pp. 119—123, [62]. 

' Cambridge and Dublin Mathematical Journal^ t. ix. (1854) pp. 63 — 67. 

» CrelU, t. L. (1866) pp. 288—299, [186]. 

c. II. 63 



498 



A MEMOm ON THE AUTOMORPHIC LINEAR TRANSFORMATION 



[153 



which stands for 

(ax + 6y -hcz )x 

+ (a'x + Vy +c'z)y 

+ (a"a? + V'y + c''z) z, 

and in which {x^ y, z) are said to be the nearer variables, and (z, y, z) the further 
variables of the bipartite. 

2. It is clear that we have 

( a , 6 , c $a?, y, ^$x, y, z) = ( a, a\ a" $x, y, z$a?, y, z) 



«'' j»'' -•" 
a , , c 



6, y, 6" 



c, c , c 



// 



and the new form on the right-band side of the equation may also be written 

(tr. ( a , 6 , c ) $x, y, z^x, y, z), 

^" !»" V 

a , , c 

that is, the two sets of variables may be interchanged, provided that the matrix is 
transposed 

3. Each set of variables may be linearly transformed: suppose that the substitu- 
tions are 

(«, y, -8^) = ( i , wi , n $a?^, y,, z,) 



and 



(X, y. z) = ( 1 . 1' . 1" $^,. y,. O- 



// 



m, m, m 
n , n , n 



Then first substituting for {x, y, z) their values in terms of (a?^, y^, z^\ the bipartite 

becomes 

( ( a , 6 , c 5^ Z , m , n ) $a?„ y„ ^,$x, y, z); 



a", 5", c" 






represent for a moment this expression by 

( A , B , C $a?,, y,, z,^x, y, z), 



153] 



OF A BIPARTITE QUADRIC FUNCTION. 



499 



then substituting for (x, y, z) their values in terms of (x^, y^, z^), it is easy to see 
that the expression becomes 

( ( 1 , m , n $ il , 5 , ) $/c„ y,, zjx„ y„ z,), 



r, m'', n' 



A\ R, cr 



and re-establishing the value of the auxiliary matrix, we obtain, as the final result of 
the substitutions, 

( a , 5 , c $a?, y, «$x, y, z) = (( 1 , m , n ISa ,h , c \l , m , n ) $a?,, y,, ^^$x,, y,, zj, 



_// L/' >." 



r, m', n' 
r, m", n' 



a', 6', c' 

_// L// -// 

a , , c 






that is, the matrix of the transformed bipartite is obtained by compounding in order, 
first or furthest the transposed matrix of substitution of the further variables, next the 
matrix of the bipartite, and last or nearest the matrix of substitution of the nearer 
variablea 

4. Suppose now that it is required to find the automorphic linear transformation 

of the bipartite 

( a , 6 , c\ X, y, z^, y, z), 

a' , V y c' 

^" u* V 

a , , c 



or as it will henceforward for shortness be written, 

(n$a?, y, ^$x, y, z); 

this may be eflfected by a method precisely similar to that employed by M. Hermite 
for an ordinary quadric. For this purpose write 

a? + a?^ = 2f, y + y, = 2i7, z + ^, = 2f, 
x + x, = 2B, y + y, = 2H, z + z^ = 2Z, 

or, as these equations may be represented, 

(a?+a:„ y-)ry,, z + -f,)= 2(f , 17 , f ), 

(x + x„ y + y,, z + z^)=2(H, H, Z); 
then we ought to have 

(n$2f-a?, 217 -y, 2f-.^][2a-x, 2H-y, 2Z - z) = (n][a;, y, -erjx, y, z). 

5. The left-hand side is 
4(ft$f, ^, ?$H, H, Z)-2(n$a?, y, ^$B, H, Z)-2(n$f 17, ?$x, y, z) + n$^, y, ^$x, y, z), 
and the equation becomes 

2(n$f 1,. r$B. H. Z)-(fl$a:, y, r$S. H. Z)-(ft$f. ,,. r$x, y. z)=:0, 

63—2 



500 A MEMOIR ON THE AUTOMORPHIC LINEAR TRANSFORMATION [153 

or as it may be written, 

("$f , V. ?$B, H, Z) -. (a^x, y, ^$B, H, Z) 



. y. z)) 



-X, H-y. Z-z){ ' 

-X. v-y. ?-«$a. H, Z)]^ 

-X, H-y. Z-z$f. ,. ?)) ' 



+ (n$f. V. ?$B. H, z) - (n$f, ,, r$x 

or acfain, 

^' (n$f-.. ,-y, ?-.$B. H. Z) 

+ («$?. I?. ?$B 

or what is the same thing, 

(ft$f-^, v-y. ?-^$H, H, Z) 

+ (tr. njB 
and it is easy to see that the equation will be satisfied by writing 

(tr.n$H-x, H-y, Z-z) = -(tr.T$B, H, Z). 
where T is any arbitrary matrix. In fact we have then 

( n$f -X, V -y, K -^$B. H, Z)= ( T$f , , , r$B, H. Z), 

(tr.n$S-x. H-y, Z-z$f . i, , r) = -(tr.T$H, H, Z$f , ,,. f) 

( T$f , ^ , ?$B. H, Z), 

and the sum of the two terms consequently vanishe& 

6. The equation 

gives 

(n-T$f, ,, K)=mj'.y.'). 

and we then have 

In fact the two equations give 

or what is the same thing, 

which is the equation assumed as the definition of (^, 17, ^); and conversely, this 
equation, combined with either of the two equations, gives the other of them. 

7. We have consequently 

(^, y, z) = (n-^ (ft -. T)$f , 17. ?), 

(f, 17, ?) = ((" + T)-^"$^., y.»0» 

and thence 

(X. y, *)=(n->(n-T)(n+T)->n$^„ y„ o. 



153] OF A BIPARTITE QUADRIC FUNCTION. 501 

8. But in like maimer the equation 

(tr.n$H-x, H-y, Z - z) = - (tr. T$B, H, Z) 
gives 



(tr. fl + T$E, H, Z) = (tr. n$x, y, z), 

and we then obtain 

(tr.fl^TJB, H, Z) = (tr.ft$x„ y,, z,). 

9. In tact these equations give 

(tr.2n$B, H, Z) = (tr.fl$x + x„ y + y„ z + z^), 
or 

2(a. H, Z) = (x + x,, y + y„ z + z,); 

and conversely, this equation, combined with either of the two equations, gives the other 
of them. We have then 



(x, y , z) = ((tr.ft)-ntr.fl + T$H, H, Z), 



(B, H, Z) = ((tr.n-T)-^ tr. n$x,, y,, z> 

and thence 

(X, y, z) = ((tr. ft)-» (tr. flTT)(tr. fl-T)-Hr. ftjx,, y„ z,). 

10. Hence, recapitulating, we have the following theorem for the automorphic linear 
transformation of the bipartite 

(n$a?, y, z^x, y, z), 
viz. T being an arbitrary matrix, if 

(x, y, ^)=:(fl-nn-T)(fl + T)-^n$a;,, y„ O, 



(x, y, z) = ((tr.n)-Mtr. n + T)(tr. fl-T)-»tr. n$x„ y„ z,), 
then 

(ft$a?, y, ^$x, y, z) = (n$a?,, y„ s,$x„ y„ z), 

which is the theorem in question. 

11. I have thought it worth while to preserve the foregoing investigation, but 
the most simple demonstration is the verification d posteriori by the actual substitution 
of the transformed values of (x, y, z), (x, y, z). To effect this, recollecting that in general 
tr. (il~*) = (tr. -4)"* and tr. ABCD ^ tr, D. tr. C, tr. B. tr. A, the transposed matrix of 
substitution for the further variables is 

ft (ft - T)-Hn + T) n-^ 

and compounding this with the matrix ft of the bipartite, and the matrix 

ft-Mft-T)(ft+T)-^ft 



502 A MEMOIR ON THE AUTOMOBPHIC LINEAR TRANSFORMATION [l53 

of substitution for the nearer variables, the theorem will be verified if the result k 
equal to the matrix fl of the bipartite ; that is, we ought to have 

or what is the same thing, 

11(1:1 - T)-Hft + T) n-Hfl - T)(ft + T)-^fl = n ; 

this is successively reducible to 

(n -h T)ft->(n - T) = (ft - T)ft-'(ft + T), 
ft-^ft + T)ft-»(ft - T) = ft-^ft - T)ft-'(ft + T), 
(1 + ft-'T)(l - ft-^T) =(l-.ft-*T)(l + fl-^T), 

which is a mere identity, and the theorem is thus shown to be true. 

12. It is to be observed that, in the general theorem, the transformations or matrices 
of substitution for the two sets of variables respectively are not identical, but it may 
be required that this shall be so. Consider first the case where the matrix ft is 
sjnumetrical, the necessary condition is that the matrix T shall be skew symmetrical ; 
in fact we have then 

tr. ft = ft, tr. T=-T, 

and the transformations become 

(x, y, ^) = (ft-Hft-T)(ft+T)-»ft$a:,, y„ z,\ 
(X, y, z) = (ft-^(ft - T)(ft + T)-^ft$x„ y„ z,), 

which are identical. We may in this case suppose that the two sets of variables 
become equal, and we have then the theorem for the automorphic linear transformation 
of the ordinary quadric 

(ft$a?, y, zy, 

viz. T being a skew sjnumetrical matrix, if 

(x, y, ^) = (ft-^n - T)(ft + T)-^ft$a?„ y„ z,), 
then 

(ft$a?, y, zy = (ft$a;„ y„ z,y, 

13. Next, if the matrix ft be skew symmetrical, the condition is that the matrix 
T shall be sjnumetrical ; we have in this case tr. ft = — ft, tr. T = T, and the four factors 
in the matrix of substitution for (x, y, z) are respectively — ft~*, — (ft — T), — (ft + T)-* 
and —ft, and such matrix of substitution becomes therefore, as before, identical with 
that for (a?, y, z)\ we have therefore the following theorem for the automorphic linear 
transformation of a skew symmetrical bipartite 

(ft$a?, y, «$x, y, z), 



153] OF A BIPARTITE QUADRIC FUNCTION. 503 

when the transformations for the two sets of variables are identical, viz. T being any 
symmetrical matrix, if 

{X. y. ^) = (fi-'(n-T)(n + T)-'n$«,. y„ z). 

(X, y, z) = (n->(fi-T)(n + T)->ft$x„ y„ z,). 

then 

(ft$a?, y, £$x, y, z)=(ft$aj,, y„ z,$x,, y„ z,). 

14. Lastly, in the general case where the matrix H is anything whatever, the 
condition is 

fl-^T = - (tr. n)-* tr. T 

for assuming this equation, then first 



n-^n - T) = (tr. n)-Htr. fn- t), 

and in like manner 



Qr\a + T) = (tr. n)-*(tr. n - T). 
But we have 



1 = (tr. n)-i(tr. n - T) (tr. n - T)-» tr. n, 
and therefore, secondly, 

(ft + T)-* ft = (tr. ft - T)-* tr. ft ; 
and thence 



ft-^n - T)(ft + T)-»ft = (tr. ft)-» (tr. ft + T) (tr. ft - T)-Hr. ft, 
or the two transformations are identical. 

15. To further develope this result, let ft"* be expressed as the sum of a 
symmetrical matrix Q^ and a skew symmetrical matrix Q^, and let T be expressed in 
like manner as the sum of a symmetrical matrix T^ and a skew symmetrical matrix 
T^. We have then 

(tr.ft)-* = tr.(ft-0 = eo-e., 
T =T, + T„ 

tr. T =To-T„ 

and the condition, ft"* T = — (tr. ft)-* tr. T, becomes 

that is, 

<2.T, + Q,T, = 0, 

and we have 



504 



A MEMOIR ON THE AUTOMORPHIC LIXEAB TRANSFORBCAHON 



[153 



or as we may, write it. 



and thence 



To = - (i{n-» + tr. ft-»})-»(J{ft-^ - tr. n-»})T,. 

T = - (i{fl-^ + tr. ft-»})-Ki{«-* - tr. fl-»})T, + T,. 



where T^ is an arbitrary skew sjnmmetrical matrix. 



16. This includes the before-mentioned special cases; first, if fi is symmetrical, 
then we have simply T^T^, an arbitrary skew symmetrical matrix, which is right 
Next, if fl is skew symmetrical, then T = — 0-*n~*T^ + T^, which can only be finite 
for T^ = 0, that is, we have T = -O-*n"*0, and (the first part of T being always 
sjnumetrical) this represents an arbitrary s}'mmetrical matrix. The mode in which this 
happens will be best seen by an exampla Suppose 

fl-i = ( A , jy+i'), tr. ft-»=( A , H^v\ 



and write 



then we have 



T, = ( 0, 6), 
-0,0 

r=-{A, ff)->( 0. !.)( 0, e)+( 0. 0) 

H, b\ -i;, o! I-^. 0' 1-^, 



v0 



±jn{-B, H) + ( 0, 0) 



AB-H 



H, -A \-0, 



_ ( vB0 



AB-H* 



-vH0 ^^) + ( 



-pH0 
AB-H 



,-^. 



AB-H' 

vA0 
AB-H* 



0, 0) 
0, 



When n is skew symmetrical. A, B, H vanish; but since their ratios remain arbitrary, 
we may write kA, kB, kH for A, B, H, and assume ultimately « = 0. Writing k0 
in the place of 0, and then putting k= 0, the matrix becomes 

( vB0 



-vB0 ) 
AB-H*' AB-H* 



-vH0 



vA0 



AB-H*' AB-H* 

which, inasmuch aa A -.0, B : 0, and C : remain arbitrary, represents, as it should do, 
an arbitrary symmetrical matrix. 



153] OF A BIPARTITE QUADRIC FUNCTION. 505 

17. Hence, finally, we have the foUoMring Theorem for the automorphic linear 
transformation of the bipartite qnadric, 

(ft$a?, y, z^x, y, z), 

when the two transformations are identical, viz. if T^ be a skew symmetrical matrix, 
and if 

T = - (i{n-^ + tr. fl-^})(J{n-^ - tr. fl-^})T, + T, ; 
then if 

(X. y, «) = (fl->(n-T)(n + T)->n$ar„ y,. z), 

(X, y, z) = (fl-> (n - T) (ft + T)-> n$x„ y,. z,); 

we have 

(n$a?, y, ^$x, y, z) = (n$a?„ y„ z,^x^, y„ z,); 

and in particular. 

If n is a sjnnmetrical matrix, then T is an arbitrary skew sjrmmetrical matrix ; 
If n is a skew symmetrical matrix, then T is an arbitrary symmetrical matrix. 



c. n. 



64 



506 



[154 



154. 



SUPPLEMENTARY RESEARCHES ON THE PARTITION OF 

NUMBERS. 



[From the PhilosophiccU Transactions of the Royal Society of London^ vol. cxlviil for 
the year 1868, pp. 47—52. Received March 19,— Read June 18, 1857.] 

The general formula given at the conclusion of my memoir, "Researches on the 
Partition of Numbers "(0> ^ somewhat diflferent ftom the corresponding formula of 
Professor Sylvester', and leads more directly to the actual expression for the number of 
partitions, in the form made use of in my memoir; to complete my former researches, 
I propose to explain the mode of obtaining from the formula the expression for the 
number of partitions. 

The formula referred to is as follows, viz. if ^ be a rational fraction, the denomi- 

fx 

nator of which is made up of &ctors (the same or different) of the form 1—^, and 
if a is a divisor of one or more of the indices m, and k is the number of indices of 
which it is a divisor, then 






where 






1 Philotophical Transactions, torn, oxlvi. (1S56) p. 127, [140]. 

' Professor Sylvester's researches are published in the Quarterly Mathematical Journal, torn. i. [1857, 
pp. 141 — 152]; there are some numerioal errors in his value of P (1, 2, 8, 4, 5, 6) q. 



154] SUPPLEMENTARY RESEARCHES ON THE PARTITION OP NUMBERS. 507 

in which formula [1 — of*] denotes the irreducible &ctor of 1 — o^, that is, the fS^^tor 
which equated to zero gives the prime roots, and /d is a root of the equation 
[1 — a?*] = ; the summation of course extends to all the roots of the equation. The 
index 8 extends from « = 1 to 8=^k; and we have then the portion of the fraction 
depending on the denominator [1— «"]. In the partition of numbers, we have ^ = 1, 
and the formula becomes therefore 



{ 



7-1 ='... + rT7^i^('^'y~'S^^^■^ 

/«j[,_«.j n(«-i)^ p-x 



where 



Uis-'l)^ " [l^a^y 



We may write 



vp = coeff. - in i^^ ^. .. . 
^^ t Ape-') 

/a? = n (1 - ««), 



where m has a given series of values the same or diflferent. The indices not divisible 
by a may be represented by m, the other indices by ap, we have then 

where the number of indices ap is equal to k. Hence 

f(pe^) = n (1 - p^'er^) H (1 - p^er^p^y, 
or since /> is a root of [! — «:*] = 0, and therefore />* = 1, we have 

fipe^) = n (1 - p^e-^) n (1 - er^J^)\ 

and it may be remarked that if n = i/ (mod. a), where v<a^ then instead of p** we 
may write p", a change which may be made at once, or at the end of the process of 
development. 

We have consequently to find 

>^ = ^^^i^^^'n(i-.pne-on(i-e-i^)- 

The development of a fS^^tor ^ ^^ is at once deduced from that of . __ . , and is 

a series of positive powers of t The development of a factor | _^^ap< ^^ deduced from 

1 . 1 * 

that of ;j — -nj, and contains a term involving -. Hence we have 

n (1 -p'e-"*) n (1 - e^) =""*-* ««'^^-*-"F5-'^'*-?'^'*'''*'^' 

and thence 

XP = P^-*- 

64—2 



508 SUPPLEMENTABT BESEABCHES ON THE PARTITION OF NUMBEBa [154 

The actual development, when k is small (for instance k^\ or £ = 2), is most readily 
obtained by developing each &ctor separately and taking the product. To do this we 
have 

where by a general theorem for the expansion of any function of e*, the coefficient 
of ^-^ is 

=ty 1 (K 

n/i-ca+A)"^ 






n/ vi-c • (i-c)» — • (i-cy+^ 

<where as usual A(V = l>'-0/ A«0/ = 2/-2 . l-^ + CK, &c.) and 

1 111. 1^. l«p 



l-6-« e"^2 12 720 30240 

where, except the constant term, the series contains odd powers only and the coef- 

l—y^^Bf .111 

ficient of t^'-^ is n9/' ^ ^*' ^^' ^^'" ^^^^^^^8 *^® series ^, ^txi tS"- ^^ Bernoulli's 

numbers. 

But when k is larger, it is convenient to obtain the development of the fraction 
firom that of the logarithm, the logarithm of the fraction being equal to the sum of 
the logarithms of the simple factors, and these being found by means of the formulae 

The fraction is thus expressed in the form 

n(i-.p«)n(ap)f* 

and by developing the exponential we obtain, as before, the series commencing with 

Besuming now the formula 

XP = P^-*> 

which gives p^ as a function of p, we have 

Ox 



[1 - a^] p - a? ' 



154] SUPPLEMENTARY RESEARCHES ON THE PARTITION OF NUMBERS. 509 

but this equation gives 

and we have 

[l-aj«] = (a?-p)(a?-p««)...(a?-.p«a), 

if 1, Oj, ... a. are the integers less than a and prime to it (a is of course the degree 
of [1 - ic*]). Hence 

and therefore 

or putting for yp its value 

^p = - p« n (1 - p«'-0 A^, 

where a is the degree of [I— a^] and o^ denotes in succession the integers (exclusive 

of unity) less than a and prime to it. The function on the right hand, by means of 

the equation [I— p*] = 0, may be reduced to an integral function of p of the degree 

a — 1, and then by simply changing p into x we have the required function 0x, The 

0x 
fraction ^i — I5T ^^° then by multiplication of the terms by the proper factor be 

reduced to a fraction with the denominator 1 — a;*, and the coefficients of the numerator 
of this fraction are the coefficients of the corresponding prime circulator ( ) per a^. 

Thus, let it be required to find the terms depending on the denominator [1 — a:*] in 



(l-a?)(l-.a:«)(l-a;»)(l-a?*)(l-a?»)(l-a;«)' 



these are 



where 



p—x p—x 






and 



fipe-') (1 - per") (1 - p»6-^) (1 - p'ir^) (1 - p»r^) (1 - O (1 - e-^) 



510 SUFPLEMEMTABT RESEARCHES ON THE PARTITIOK OF NUMBEBS. [15^ 

where it is easy to see that 

1 1 



A^ = 



18il-p)(l-p')(l-p^)(l-p»)' 



A 1 fl 1 f p ip' 4,p* ^p^ \] 

^-'~(l-p)(l-p')(l-f^){l-p')\* 18\l-p^l-f^'^l-p*^l-p^)]' 

and we have 

e^ = -p*(l-p)A^, 

^,p = -p»(l-p)4_,. 

But [1 — p*] = 1 + p + p* = 0. Hence p* = 1, and therefore 

(1 -p)(l -p'Xl -p*)(l -p')-(l -p)«(l -p')'=9. 

Hence 

<'sP = -i^p'(l-p) = 4(l-p')=l^(2+p), 

whence 

and the partial fraction is 

1 2-hx 



1621+a? + a'»' 

which is 

""162 1-a^ ' 
and gives rise to the prime circulator t^ (2, —1. — l)pcr 3,. 
The reduction 0ip is somewhat less simple; we have 

= ^(l-p')(51-10p-14p0 
= ^ (61 + 4p - 65p') ; 



154] SUPPLEMENTARY BESEABCHES ON THE PARTITION OF NUMBERS. 511 



hence finally 



^^P = m^-''^ '^ ^^f'^' ^i« = 3^(42 + 23a:); 



and the partial fraction ia 



which is 



J_ . 42 + 23a! 
S2* '1+x + a/" 



1 - 42-19a;-23g' 

324 '^^ r^ 



and gives rise to the prime circulator ssi. ? (42, — 19, — 23) per 3 



324 
The part depending on the denominator 1 — a; is 



^-' +1^0.^^ + A (^e.)* ^ 



l-x 1 "l-a; ■ 1.2 



where 



l-x 
1 



...+ 



1.2.3.4.5 



(xd,y 



A. 



l-x' 



We have here 



(1 - O (1 - e-*) (1 - «-") (1 - «-^) (1 - e-«) (1 - e-«) 

A i . A ^ ,1 1 P 

= il-. t; + -d_, - ... + j4_, - +&C. 



logi-3^ = -log« + 2<-24^ + 2880^-'^-' 



and thence the fraction is 



, 21, 91„.4S5 



720 f 



which is equal to 



720 «• 



21 , . 441 ^ . 3087 ^ . 64827 ^ . 1361367 ^ , 
t + -g-t + ^g- «• + -128" ^ + 1280 "''••• 



!0 «• /, 21 

/, 91^ 8281^ \ 



11 7 1 77 1 245 1 . 43981 1 . 199677 1 



720 f*^ 480 «• 1080 «« 1152 f 103680 1? 345600 < 
and consequently the partial fractions are 

(ae.)» , + ^^^5a (^'Y ^—- + cJ5a («®«)' 1 1 + ooAi (a^*)* 



86400 



1 - a; ■ 11520 



l-x 6480 



1-a; ' 2304 



103680 
from which the non-circulating part is at once obtained. 



. 43981 ,_^ , 1 ^ 

+ ,,>»^»^ (aW») :; + 



l-x 
199577 1 



1 - a;^ 345600 l-a;' 



512 SUPPLEMENTARY RESEARCHES ON THE PARTITION OF NUMBERS. [154 

The complete expression for the number of partitions is P (1, 2, 3, 4, 5, 6) q = 
^QgggQQ (IV + 63(V + 12303» + 1102503» + 4398109 + 698731) 

+ ^ (69' + 126g + 581)(l. 1) per 2, 

+ Jg2? (2, - 1, - 1) per 3, 

+ 3^4 (42, -19. -23) per 3, 

+ 32 (1, 1, - 1, - 1) per 4, 

+ ^ (2, 1, 0, -1.-2) per 5, 

+ ^ ..(2. 1,-1, -2, -1,1) per 6^ 



155] 



518 



155. 



A FOURTH MEMOIR UPON QUANTICS. 

[From the Philosophical Transactions of the Royal Society of London, vol. cxlviil fir 
the year 1858, pp. 415—427. Received February 11,— Read March 18, 1858.] 

The object of the present memoir is the fiirther development of the theory of binary 
quantics ; it should therefore have preceded so much of my third memoir, t. 147 (1857), 
p. 627, [144], as relates to ternary quadrics and cubics. The paragraphs are numbered 
continuously with those of the former memoira The* first three paragraphs, No& 62 to 64, 
relate to quantics of the general form (♦$a?, y, ...)^i and they are intended to complete 
the series of definitions and explanations given in Nos. 54 to 61 of my third memoir; 
Nos. 68 to 71, although introduced in reference to binary quantics, relate or may be 
considered as relating to quantics of the like general form. But with these exceptions 
the memoir relates to binary quantics of any order whatever: viz. Nos. 65 to 80 relate 
to the covariants and invariants of the degrees 2, 3 and 4; Nos. 81 and 82 (which are 
introduced somewhat parenthetically) contain the explanation of a process for the 
calculation of the invariant called the Discriminant; Nos. 83 to 85 contain the definitions 
of the Catalecticant, the Lambdaic and the Canonisant, which are functions occurring in 
Professor Sylvester's theory of the reduction of a binary quantic to its canonical form ; 
and Nos. 86 to 91 contain the definitions of certain covariants or other derivatives 
connected with Bezout's abbreviated method of elimination, due for the most part to 
Professor Sylvester, and which are called Bezoutiants, Cobezoutiants, &a I have not in 
the present memoir in any wise considered the theories to which the catalecticant, &c. 
and the last-mentioned other covariants and derivatives relate; the design is to point 
out and precisely define the different covariants or other derivatives which have hitherto 
presented themselves in theories relating to binary quantics, and so to complete, as far 
as may be, the explanation of the terminology of this part of the subject. 

62. If we consider a quantic 

(a, 6, ...$a?, y, ...)•" 
c. II. 65 



514 A FOURTH MEMOIR UPON QU ANTICS. [l55 

and an adjoint linear form, the operative quantic 

(a, 6, ...$3|, 9,,...)'*, 

or more generally the operative quantic obtained by replacing in any covariant of the 
given quantic the facients (a?, y, ...) by the symbols of differentiation (9^. 9,,...) (which 
operative quantic is, so to speak, a contravariant operator), may be termed the Pro- 
vector ; and the Provector operating upon any contravariant gives rise to a contra- 
variant, which may of course be an invariant. Any such contravariant, or rather such 
contravariant considered as so generated, may be termed a Provectant; and in like 
manner the operative quantic obtained by replacing in any contravariant of the given 
quantic the facients (f , 17, ...) by the sjrmbols of differentiation (9jb, 9y, ...) (which operative 
quantic is a covariant operator), is termed the Cantraprovector ; and the contraprovector 
operating upon any covariant gives rise to a covariant, which may of course be an 
invariant. Any such covariant, or rather such covariant considered as so generated, 
may be termed a Contraprovectant 

In the case of a binary quantic, 

(a, 6, ...][a?, y)*", 

the two theorems coalesce together, and we may say that the operative quantic 

(a, 6, ...$9y,-9«)"*, 

or more generally the operative quantic obtained by replacing in any covariant of the 
given quantic the facients (a?, y) by the symbols of differentiation (9y, — 9,) (which is 
in this case a covariant operator), may be termed the Provector. And the Provector 
operating on any covariant gives a covariant (which as before may be an invariant), 
and which considered as so generated may be termed the Provectant. 

63. But there is another allied theory. If in the quantic itself or in any covariant 
we replace the facients (x, y, ...) by the first derived functions (9fP, 9^, ...) of any con- 
travariant P of the quantic, we have a new function which will be a contravariant of 
the quantic. In particular, if in the quantic itself we replace the &cient8 (a?, y, ...) by 
the first derived functions (9fP, 9^,...) of the Reciprocant, then the result will contain 
as a &ctor the Reciprocant, and the other factor will be also a contravariant. And 
similarly, if in any contravariant we replace the facients (f, 17,...) by the first derived 
functions (9«TF, dyW,..,) of any covariant W (which may be the quantic itself) of the 
quantic U, we have a new function which will be a covariant of the quantia And in 
particular if in the Reciprocant we replace the facients (f, 17, ...) by the first derived 
functions (dxU, dyU, ...) of the quantic, the result will contain IT as a fisu;tor, and the 
other factor will be also a covariant. In the case of a binary quantic (a» 6, ...$«, y)* 
the two theorems coalesce and we have the following theorem, viz. if in the quantic 
U or in any covariant the facients (x, y) are replaced by the first derived functions 
(^yWy — 9«TF) of a covariant TT, the result will be a covariant ; and if in the quantic 



155] A FOURTH MEMOIR UPON QU ANTICS. 515 

U the facients (a?, y) are replaced by the first derived functions (9yJ7, —dxU). of the 
quantic, the residt will contain ^ as a factor^ and the other fiEU)tor will be also a 
covariant. 

Without defining more precisely, we may say that the function obtained by replacing 
as above the facients of a covariant or contravariant by the first derived functions of a 
contravariant or covariant is a Transmutant of the first-mentioned covariant or contra- 
variant. 

64. Imagine any two quantics of the same order, for instance, the two quantics 

F==(a', 6',...$a?, y ...)'", 

then any quantic such as \U-hfiV may be termed an Intermediate of the two quantics; 
and a covariant of \U+fiV, if in such covariant we treat \, /i as facients, will be a 
quantic of the form 

where the coeflScients (A, B, ... R, A") will be covariants of the quantics U, F, viz. A 

will be a covariant of the quantic U alone ; J." will be the same covariant of the quantic 

V alone, and the other coefficients (which in reference to A, A^ may be termed the 

Connectives) will be covariants of the two quantics; and any coefficient may be obtained 

from the one which precedes it by operating on such preceding coefficient with the 

combinantive operator 

a'3a+6'96 + ..., 

or fi-om the one which succeeds it by operating on such succeeding coefficient with the 
combinantive operator 

ad a' + idb' + • • . > 

the result being divided by a numerical coefficient which is greater by unity than 

the index of fi or (as the case may be) \ in the term corresponding to the coefficient 

operated upon. It may be added, that any invariant in regard to the fiaicients (X, fi) 

of the quantic 

(A, B, ... R, A'^\ fiY 

is not only a covariant, but it is also a combinant of the two quantics 17, F. 

As an example, suppose the quantics IT, F are the quadrics 

(a, 6, c\x, yy and (a\ h\ c'\x, yf, 

then the quadrinvariant of 

XlT+ZiF is (Xa + fta'XXc + /ic') - (X6 + /iftO*. 
which is equal to 

{ac-}?, ac'-2hV + ca\ a'c'-6'«5x, /*)», 

and oc' — 266' + ca' is the connective of the two discriminants oc — 6* and aV — 6'*. 

65—2 



516 A FOURTH MEMOIB UPON QUANTICS. [l55 

65. The law of reciprocity for the number of the invariants of a binary quantic\ 
leads at once to the theorems in regard to the number of the quadrinvariantSy cabin- 
variants and quartinvariants of a binary qmmtic of a given degree, first obtained by 
the method in the second part of my original memoir*. Thus a quadric has only a 
single invariant, which is of the degree 2 ; hence, by the law of reciprocity, the number 
of quadrinvariants of a quantic of the order m is equal to the number of ways in which 
m can be made up with the part 2, which is of course unity or zero, according as 
m is even or odd. And we conclude that 

The quadrinvariant exists only for quantics of an even order, and for each such 
quantic there is one, and only one, quadrinvariant. 

66. Again, a cubic has only one invariant, which is of the degree 4, and the 
number of cubinvariants of a quantic of the degree m is equal to the number of 
ways in which m can be made up with the part 4. Hence 

A cubinvariant only exists for quantics of an evenly even order, and for each 
such quantic there is one, and only one, cubinvariant. 

67. But a quartic has two invariants, which are of the degrees 2 and 3 respectively, 
and the number of quartinvariants of a quantic of the degree m is equal to the number 
of ways in which m can be made up with the parts 2 and 3. When m is even, 
there is of course a quartinvariant which is the square of the quadrinvariant, and which, 
if we attend only to the irreducible quartinvariants, must be excluded from consideration. 
The preceding number must therefore, when m is even, be diminished by unity. The 
result is easily found to be 

Quartinvariants exist for a quantic of any order, even or odd, whatever, the quadric 
and the quartic alone excepted; and according as the order of the quantic is 

6(7, 6(7 + 1, 6(7 + 2, 6^ + 3, 6(7 + 4, 6(7 + 5, 
the number of quartinvariants is 

g, 9 f 9 » 5^ + 1, 9 > 5^+1- 
In particular, for the orders 

2, 3, 4, 5; 6, 7, 8, 9, 10, 11; 12, &a, 

the numbers are 

0, 1, 0, 1; 1, 1, 1, 2, 1, 2; 2, &c 
Thus the ninthic is the lowest quantic which has more them one quartinvariant. 

68. But the whole theory of the invariants or covariants of the degrees 2, 3, 4 is 
most easily treated by the method above alluded to, contained in the second part of my 
original memoir; and indeed the method appears to be the appropriate one for the 

I Introduistoiy Memoir, [189], No. 20. * Ibid. No8. 10-17. 



1 

i 



155] A FOUKTH MEMOIR UPON QUANTIC8. 517 

treatment of the theory of the invariants or covariants of any given degree whatever, 
although the application of it becomes difficult when the degree exceeds 4. I remark, 
in regard to this method, that it leads naturally, and in the first instance, to a special 
class. of the covariants of a system of quantics, viz. these covariants are linear functions 
of the derived functions of any quantic of the system. (It is hardly necessary to remark 
that the derived functions referred to are the derived functions of any order of the 
quantic with regard to the facients.) Such covariants may be termed tantipartite 
covariants; but when there are only two quantics, I use in general the term lineo-linear. 
The tantipartite covariants, while the system remains general, are a special class of 
covariants, but by particularizing the system we obtain all the covariants of the par- 
ticularized system. The ordinary case is when all the quantics of the system reduce 
themselves to one and the same quantic, and the method then gives all the covariants 
of such single quantic. And while the order of the quantic remains indefinite, the 
method gives covariants (not invariants); but by particularizing the order of the quantic 
in such manner that the derived functions become simply the coefficients of the quantic, 
the covariants become invariants: the like applies of course to a sjrstem of two or more 
quantics. 

69. To take the simplest example, in seeking for the covariants of a single quantic 

Uj we in fact have to consider two quantics U, V, An expression such as 12 IT 7" is a 
lineo-linear covariant of the two quantics; its developed expression is 

which is the Jacobian. In the particular case of two linear functions (a, 6][a?, y) and 
(of, V^x, y)y the lineo-linear covariant becomes the lineo-linear invariant ab' — a% which 
is the Jacobian of the two linear fiinctiona 

In the example we cannot descend from the two quantics U,VU> the single quantic 
U (for putting F= 17 the covariant vanishes); but this is merely accidental, as appears 

by considering a diflferent lineo-linear covariant 12* [/"F", the developed expression of 
which is 

In the particular case of two quadrics (a, 6, c$x, yf, (a\ b', (/'^x, y)\ the lineo-linear 
covariant becomes the lineo-linear invariant 

ac' - 266' + ca'. 

If we have V—U, then the lineo-linear covariant gives the quadricovariant 

d»^U.dy^U^(dJdyUy 

of the single quantic U (such quadricovariant is in &ct the Hessian) ; and if in the last- 
mentioned formula we put for U the quadric (a, 6, c][, x, yf, or what is the same thing, 
if in the expression of the lineo-linear invariant ac' — 266' + ca', we put the two quadrics 
equal to each other, we have the quadrinvariant 

ac— 6* 
of the single quadric. 



518 A FOURTH MEMOIR UPON QUANTICS. [155 

70. The lineo-lineajr invariant ah* — o!h of two linear functions may be considered as 
giving the lineo-Iinear covariant d^U .dyV—dyU . dgV of the two quantics U and F, 
and in like manner the lineo-linear invariant ac' — 2bb' + ca* may be considered as giving 
the lineo-linear covariant dx^U .dy^V—^dJdyU .dJdyV+dy^U .dx*V o{ the quantics U, V. 
And generally, any invariant whatever of a quantic or quantics of a given order or orders 
leads to a covariant of a quantic or quantics of any higher order or orders: viz. the 
coefficients of the original quantic or quantics are to be replaced by the derived functions 
of the quantic or quantics of a higher order or ordera 

71. The same thing may be seen by means of the theory of Emanants. In tact, 
consider any emanants whatever of a quantic or quantics; then, attending only to the 
facients of emanation, the emanants will constitute a system of quantics the coefficients 
of which are derived functions of the given quantic or quantics; the invariants of the 
system of emanants will be functions of the derived functions of the given quantic or 
quantics, and they will be covariants of such quantic or quantics; and we thus pass 
from the invariants of a quantic or quantics to the covariants of a quantic or quantics 
of a higher order or orders. 

72. It may be observed also, that in the case where a tantipartite invariant, when 
the several quantics are put equal to each other, does not become equal to zero, we may 
pass back from the invariant of the single quantic to the tantipartite invariant of the 
system ; thus the lineo-linear invariant ac* — 266' + ca' of two quadrics leads to the quadiin- 
variant ac — 6* of a single quantic ; and conversely, from the quadnnvariant oc — 6* of a 
single quadric, we obtain by an obvious process of derivation the expression cuf — 266' + ca' 
of the lineo-linear invariant of two quadrics This is in fact included in the more general 
theory explained. No. 64. 

73. Reverting now to binary quantics, two quantics of the same order, even or odd, 
have a lineo-linear invariant. Thus the two quadrics 

(a, 6, c^x, yy, (a\ h\ c'^a?, yf 

have (it has been seen) the lineo-linear invariant 

acf - 266' + ca* ; 

and in like manner the two cubics 

(a, 6, c, djir, y)», (a', 6', c', d'$a:, yf 

have the Uneo-Unear invariant 

ad! - 36c' + 3c6' - da\ 

which examples are sufficient to show the law: 

74. The lineo-linear invariant of two quantics of the same odd order is a combinant, 
but this is not the case with the lineo-linecur invariant of two quantics of the same even 
order. Thus the last-mentioned invariant is reduced to zero by each of the operations 



155] A FOURTH MEMOIR UPON QUANTICS. 519 

and 



but the invariant 
is by the operations 

and 

reduced respectively to 

and 



afda + b'db + c'dc + cfda ; 
ac' -266' 4- ca' 

a'da + Vdb + c'dc 
2(ac - 6^ 

2(aV-6'0. 



75. For two quantics of the same odd order^ when the quantics are put equal to 
each other, the lineo-linear invariant vanishes; but for two quantics of the same even 
order, when these are put equal to each other, we obtain the quadrinvariant of the single 
quantia Thus the quadrinvariant of the quadric (a, 6, c^x, yY is 

oc — 6*; 

and in like manner the quadrinvariant of the quartic (a, 6, c, (2, e\x, yY is 

oe - 46d + 3c». 

76. When the two quantics are the first derived functions of the same quantic 
of any odd order, the lineo-linear invariant does not vanish, but it is not an invariant 
of the single quantic. Thus the Uneo-linear invariant of 

(a, 6, c$a?, yy 

and 

(6, c, d^x, yy 

is 

(ad — 26c -f c6 = ) od — 6c, 

which is not an invariant of the cubic 

(a, 6, c, d$a?, y)". 

But for two quantics which are the first derived functions of the same quantic of 
any even order, the lineo-linear invariant is the quadrinvariant of the single quantic. 
Thus the lineo-linear invariant of 

(a, 6, c, d^x, yy 

and 

(6, c, d, e^x, yy 

is 

(a« - 36d + 3c" - d6 =) a« - 4f6(i + 3c*, 

which is the quadrinvariant of the quartic 

(a, 6, c, d, e$x, yy. 



520 A FOURTH MEMOIR UPON QUANTICS. [155 

77. I do not stop to consider the theory of the lineo-linear covariants of two 
quanties, but I derive the quadricovariants of a single quantic directly fix>m the 
quadrinvariant. Imagine a quantic of any order even or odd. Its successive even 
emanants will be in regard to the facients of emanation quantics of an even order, 
and they will each of them have a quadrinvariant, which will be a quadricovariant of 
the given quantic. The emanants in question, beginning with the second emanant, are 
(in regard to the facients of the given quantic assumed to be of the order m) of the 
orders m — 2, m — 4,... down to 1 or 0, according as m is odd or even, or writing 
successively 2p+l and 2p in the place of m, and taking the emanants in a reverse order, 
the emanants for a quantic of any odd order 2p-f 1 are of the orders 1, 3, 5... 2p — 1, 
and for a quantic of any even order 2p, they are of the orders 0, 2, 4 ... 2p— 2. The 
quadricovariants of a quantic of an odd order 2p + 1, are consequently of the orders 
2, 6, 10...4p — 2, and the quadricovariants of a quantic of an even order 2p, are of 
the orders 0, 4, 8 ... 4p — 4. We might in each case carry the series one step further, 
and consider a quadricovariant of the order 4p -f 2, or (as the case may be) 4p, which 
arises from the 0th emanant of the given quantic; such quadricovariant is, however, 
only the square of the given quantic. 

78. In the case of a quantic of an evenly even order (but in no other case) we 
have a quadricovariant of the same order with the quantic itsel£ We may in this 
case form the lineo-linear invariant of the quantic and the quadricovariant of the same 
order: such lineo-linear invariant is an invariant of the given quantic, and it is of 
the degree 8 in the coefficients, that is, it is a cubinvariant. This agrees with the 
before-mentioned theorem for the number of cubinvariants. 

79. In the case of the quartic (a, 6, c, d, e$a?, y)*, the cubinvariant is, by the 
preceding mode of generation, obtained in the form 

c(ac- 6») -4dJ(ad- 6c)-f 6cJ (oe -46d + 3c«)- 46| (be-cd) + a(ce- (?), 

which is in ^t equal to 

3 (ace -acP- b^e -f 2bcd - c») ; 

and omitting the numerical factor 3, we have the cubinvariant of the quartic 

• 

80. In the case of a quantic of any order even or odd, the quadrinvariants of the 
quadricovariants are quartinvariants of the quantic. But these quartinvariants are not 
all of them independent, and there is no obvious method grounded on the preceding 
mode of generation for obtaining the number of the independent (asyzygetic) quartin- 
variants, and thence the number of the irreducible quartinvariants of a quantic of a 
given order. 

81. I take the opportunity of giving some additional developments in relation to 
the discriminant of a quantic 

(a, 6, ...6\ a'][a?, y)'». 

To render the signification perfectly definite, it should be remarked that the discrimiuant 
contains the term a"*^^a"*^^ and that the coefficient of this term may be taken to be 



155] A FOURTH MEMOIB UPON QUANTICS. 521 

+ 1. It was noticed in the Introductory Memoir, that, by Joachimstharjs theorem, the 
discriminant, on putting a = 0, becomes divisible by 6', and that throwing out this 
factor it is to a numerical factor pris the discriminant of the quantic of the order 
(w— 1) obtained by putting a = and throwing out the factor x] and it was also 
remarked, that this theorem, combined with the general property of invariants, afforded 
a convenient method for the calculation of the discriminant of a quantic when that 
of the order immediately preceding is known. Thus let it be proposed to find the 
discriminant of the cubic 

(a, b, c, d^x, yy. 

Imagine the discriminant expanded in powers of the leading coefficient a in the form 

then this function. gu^l invariant must be reduced to zero by the operation 369a + 2cdb +dde; 
or putting for shortness V = 2<^t + dde, the operation is V + Sbda, and we have 

a^VA+aVB +V(7l 

^ = 0, 
-f a 664 -f SbBj 

and consequently 

But C is equal to 6* into the discriminant of (36, 3c, d^x, y)*, that is, its value is 
6'(126(i— 9c'), or throwing out the factor 3, we may write 

C = 46»d-36»c«; 
this gives 

£ = - ^^(- 66»cd + 246»cd- 126c*), 

or reducing 

^ = -66cd + 4c»; 
and thence 

il = - ^ (- 66d* + Uc'd - 12c»d), 

or reducing 

A=d\ 

which verifies the equation VA = 0, and the discriminant is, as we know, 

a«d« - 6a6cd -f 4ac» + 46»d - 36V. 

82. If we coiisider the quantic (a, 6, ...a$jr, 1)** as expressed in terms of the 
roots in the form a(x '-ay)(x - /3i/)..., then the discriminant (= a'""^ a *""* + &c. as 
above) is to a factor prh equal to the product of the squares of the differences of 
the roots, and the factor may be determined as follows: viz. denoting by f(a, /S, ...) 
the product of the squares of the differences of the roots, we may write 

a*"-» ? (a, 13, . . . ) = -AT (d^^ a'"*"* + &c.), 
c. II. G6 



522 



A FOUBTH MEMOIB UPON QUANTIC8. 



[155 



where ^ is a number ; and then considering the equation a;*" — 1 = 0, we have to 
determine N the equation 



But in general 



and if 



then 



or 



here 



and therefore 



but 



or 



and 



whence 



or 



and consequently 



{:(«. /9, .;.) = (-)•»-■ .V. 



<f>x = (x — a) (x — 0) ..., 
(a-/3)(a-7)... = <^'a, &c., 
?(a, /3, ...) = (-)*"*<"'-'» f«f/3...; 

(-ra/87... = -l, 
a/37...=(-)«-U, 

J^ = (-)4nr(m-i) ^,n^ 

a*"-* ?(«, /9, . ; .) = (-)*'"'"»->' m'" (a"*-» a "»-' + &c.), 



or what is the same thing, the value of the discriminant D (=a'*"*a'*~* + &c.) is 

(-)Jm(m-i)^-mam-jf(a, /8, ...)• 

It would have been allowable to define the discriminant so as that the leading term 
should be 

m 

in which case the discriminant would have constantly the same sign as the product 
of the squared differences; but I have upon the whole thought it better to make 
the leading term of the discriminant always positive. 

83. A quautic of an even order 2p has an invariant of peculiar simplicity, viz. 

the determinant the terms of which are the coefficients of the pih differential 

coefficients, or derived functions of the quantic with respect to the facients ; such 

invariant may also be considered as a tantipartite invariant of the pih emanaubi. 

Thus the sextic 

(a, 6, c, (/, e, /, flr$d?, yY 



155] A FOUBTH MEMOIR UPON QU ANTICS. 523 

has for one of its invariante, the determinant 

\ a, b, c, d , 

\ b, Cy d, e \ 

c , cf , e , f 

d^ ey f , g 

The invariant in question is termed by Professor Sylvester the Catalecticaut. 

84. Professor Sylvester also remarked, that we may from the eatalecticant form 
a function containing an indeterminate quantity X, such that the coefficients of the 
diflFerent powers of X are invariants of the quantic; thus for the sextic, the function 
in question is 

a y b ,c ,d— \ 

b , c , cf + JX, c 

, c , d - JX, e , / 

d+\ e , f y g 

where the la»v of formation is manifest; the terms in the sinister diagonal are 
modified by annexing to their numerical submultiples of X with the signs + and — 
alternately, and in which the multipliers are the reciprocals of the binomial coefficients. 
The function so obtained is termed the Lambdaie, 

85. If we consider a quantic of an odd opder, and form the eatalecticant of the 
])enultimate emanant, we have the covariant termed the Canonisant. Thus in the case 
of the quintic 

(a, 6, c, dy ey f \x, y)\ 
the canonisant is 

j ax-{-bg, bx + cy, cx + dy \ 

\ bx -{- eg, cx-^dyy dx-^-eg .\ 

' ex H- dy, dx+ «y, ex -vfg 

which is equivalent to 

a , 6 , c , d 

by C y d y € 

C , d y e y / , 

and a like transformation exists with respect to the canonisant of a quantic of any 
odd order whatever. The canonisant and the lambdaie (which includes of course the 

• 

eatalecticant) form the basis of Professor Sylvesters theory of the Canonical Forms 
of quantics of an odd and an even order respectively. 

66—2 



524 A FOURTH MEMOIR UPON QUANTIC8. [l55 

86. There is another family of covariants which remains to be noticed. Consider 
any two quantics of the same order, 

(a, 6,...$a?, y)"», 
(a, 6', ...$a?, y)~, 

and join to these a quantic of the next inferior order, 

where the coefficients {u, v, ...) are considered as indeterminate, and which may be 
spoken of as the adjoint quantic. 

Take the odd iineo-iinear covariants (viz. those which arise from the odd emanants) 
of the two quantics; the term arising from the (2i + l)th emanants is of the form 

where (-4, jB, ...) are lineo-lineai* functions of the coefficients of the two quantics. 

Take also the quadricovariants of the adjoint quantic; the term arising from the 
(2i — m)th emanant is of the form 

where {U, F, ...) are quadric functions of the indeterminate coefficients (u, v, ...). We 
may then form the quadrin variant of the two quantics of the order 2(m — 1 — 2i): 
this will be an invariant of the two quantics and the adjoint quantic, lineo-linear in 
the coefficients of the two quantics and of the degree 2 in regard to the coefficients 
(m, v, ...) of the adjoint quantic; or treating the last-mentioned coefficients as iacient«, 
the result is a lineo-linear m-ary quadric of the form 

(a, aB,...$u, r,...)«, 

viz. in this expression the coefficients J3[, J3, ... are lineo linear functions of the co- 
efficients of the two quantics. And giving to % the different admissible values, viz. 
from t = to i = ^m — 1 or ^^(m — 1) — 1, according as m is even or odd, the number 
of the functions obtained by the preceding process is Jm or ^ (m — 1), according as 
m is even or odd. The functions in question, the theory of which is altogether due 
to Professor Sylvester, are termed by him Cohezoutiants ; we may therefore say that 
a cobezoutiant is an invariant of two quantics of the same order m, and of an adjoint 
quantic of the next preceding order w — 1, viz. treating the coefficients of the adjoint 
quantic as the facients of the cobezoutiant, the cobezoutiant is an 7H-ary quadric, the 
coefficients of which are lineo-linear functions of the coefficients of the two quantics, 
and the number of the cohezoutiants is ^i or ^(m — 1), according as m is even or 
odd. 

87. If the two quantics are the differential coefficients, or first derived functions 
(with respect to the facients) of a single quantic 

(a, 6, ...$a?, y)*~. 



155] 



A FOUBTH MEMOIR UPON QUANTICS. 



525 



then we have what are termed the Cohezoutoids of the single quantic, viz. the cobe- 
zoutoid is au invariant of the single quantic of the order m, and of an adjoint quantic 
of the order (m — 2) ; and treating the coefficients of the adjoint quantic as facients, 
the cobezoutoid is an (wi — l)ary quadric, the coefficients of which are quadric functions 
of the coefficients of the given quantic. The number of the cobezoutoids is ^(m — 1) 
or ^ (m — 2), according as m is odd or even. 

88. Consider any two quantics of the same order, 

(a, . . .$a?, y)*~, (a', ...v$a?, y)*", 
and introducing the new facients (X, F), form the quotient of determinants, 



(a, ...$0?, y r, {a,.,.^x , y) 



^ , y 
X,Y 



which is obviously an integral function of the order {m — 1) in each set of facients 
separately, and lineo-linear in the coefficients of the two quantics; for instance, if the 
two quantics are 

(a, 6, c, d^x, yy, 

{a\ h\ c\ d'^x, yY, 
the quotient in question may be written 

( 3 (at' - a'b\ 3 [ac' - a'c) , ad' - a!d \x, y)« {X, Y)\ 

3 {ac' - a'c), ad' - a'd + 9 Q>c' - 6'c), 3 {bd' - b'd) 
ad'-a'd, Sibd'-b'd) , 3 {cd' ^ c'd) 

The function so obtained may be termed the Bezoutic Emanant of the two quantics. 

89. The notion of such function was in fact suggested to me by Bezout s abbre- 
viated process of elimination, viz. the two quantics of the order m being put equal to 
zero, the process leads to (w — 1) equations each of the order (m — 1): these equations 
are nothing else 'than the equations obtained by equstting to zero the coefficients of 
the different terras of the series {X, F)*^* in the Bezoutic emanant, and the result 
of the elimination is consequently obtained by equating to zero the determinant 
formed with the matrix which enters into the .expression of the Bezoutic emanant. 
In other words, this determinant is the Resultant of the two quaiitic& Thus lAie resultant 
of the last-mentioned two cubics is the determinant 



3(a6'-ai), 3(ac'-a'c) , ad'-a'd 

3(ac'-a'c), ad! - a'd + 9 (be' - b'c), 3(6d'-6'rf) 

ad'-a'd, 3bd'-b'd , 3(cd'-c'rf) 



526 



A FOURTH MEMOIR UPON QUANTICS. 



[155 



90. If the two qualities are the dififerential coeflScients or first derived functions 
(with respect to the facients) of a single qiiantic of the order m, then we hare in 
like manner the Bezoutoidal Emanant of the single quantic; this is a function of the 
order (m — 2) in each set of facients, and the coefficients whereof are quadric functions 
of the coeflBcients of the single quantic. Thus the Bezoutoidal emanant of the quartic 

(a, 6, c, d, e\x, yY 



IS 



( 3(ac-6»), 3 (ad -be) , ae-bd 5^, y)^(X, Yf 
3(ad-6c), ae + Sbd-Qd', S(b€-cd) 
ae — bd, 3(b€ — cd) , 3 (ce - d* ) , 



and of course the determinant formed with the matrix which enters into the expression 
of the Bezoutoidal Emanant, is the discriminant of the single quantic. 

91. Professor Sylvester forms with the matrix of the Bezoutic emanant and a set 
of m facients («i, v, ...) an m-ary quadric function, which he terms the Bezoutiant 
Thus the Bezoutiant of the before- mentioned two cubics is 

i 3 {aV - a'6), 3 {ac' - a'c) , ad' - a'd ^w, v, wf ; 

i 3(ac'-ac), ad' - a'd 4- 9 (6c' - 6'c), 3(6d'-6'd) 
I ad' - a'd , 36d' - 6'd ,3 (cd' - c'd) , 

and in like manner with the Bezoutoidal emanant of the single quantic of the order wi 
and a set of (m — 1) new facients (w, v, ...), an (m— l)ary quadric function, which he 
terms the Bezoutoid. Thus the Bezoutoid of the before-mentioned quartic is 

( 3 (ac — 6'), 3 (ad — be) , ae — bd $w, v, w)\ 
3 (ad - 6c), ae 4- 86d - 9c», 3 {be- cd) 
ae —bd, 3{be — cd), 3 (ce — d*) 

To him also is due the important theorem, that the Bezoutiant is an invariant of 
the two quantics of the order m and of the adjoint quantic (u, v, ...$y, — a:)**~S being in 
fact a linear function with mere numerical coefficients of the invariants called Cobe- 
zoutiants, and in like manner that the Bezoutoid is an invariant of the single quantic 
of the order m and of the adjoint quantic {u, v, ...$y, — a?)"*"*, being a linear function 
with mere numerical coefficients of the invariants called Cobezoutoids. 

The modes of generation of a covariant are infinite in number, and it is to be 
anticipated that, as new theories arise, there will be frequent occasion to consider new 
processes of derivation, and to single out and to define and give names to new co- 
variants. But I have now, I think, established the greater part by far of the definitions 
which are for the present necessary. 



1561 



527 



156. 



A FIFTH MEMOIR UPON QUANTICS. 



[From the Philosophical Transactions of the Royal Society of London, vol. cxLViii. for 
the year 1858, pp. 429—460. Received February 11,— Read March 18, 1858.] 

The present memoir was originally intended to contain a development of the 
theories of the covariants of certain binary quantics, viz. the quadric, the cubic, and 
the. quartic; but as regards the theories of the cubic and the quartic, it was found 
necessary to consider the case of two or more quadrics, and I have therefore com- 
prised such systems of two or more quadrics, and the resulting theories of the 
harmonic relation and of involution, in the subject of the memoir; and although the 
theory of homography or of the anharmonic relation belongs rather to the subject of 
bipartite binary cjuadrics, yet from its connexion with the theories just referred to, it 
is also considered in the memoir. The paragraphs are numbered continuously with 
those of my former memoirs on the subject : Nos. 92 to 95 relate to a single quadric ; 
Nos. 96 to 114 to two or more quadrics, and the theories above referred to; Nos. 
115 to 127 to the cubic, and Nos. 128 to 145 to the quartic. The several quantics 
are considered as expressed not only in terms of the coefficients, but also in terms of 
the roots, — and I consider the question of the determination of their linear factors, — 
a question, in effect, identical with that of the solution of a quadric, cubic, or 
biquadratic equation. The expression for the linear factor of a quadric is deduced 
from a well-known formula; those for the linear factors of a cubic and a quartic 
were *fir8t given in my " Note ' sur les Covariants d'une fonction quadratique, cubique 
ou biquadratique k deux indc^termin^es," Crelle, vol. L. (.1855), pp. 285 — 287, [135]. It is 
remarkable that they are in one point of yiew more simple than the expression for 
the linear factor of a quadric. 

92. In the case of a quadric the expressions considered are 

(a, 6, cjx, y)\ (1) 

ac-6« , . (2) 



528 A FIFTH MEMOIR UPON QU ANTICS. [l56 

where (1) is the quadric, and (2) is the discriminant, which is also the quadrin variant, 
catalecticant, and Hessian. 

And where it is convenient to do so, I write 

(2) = D. 

93. We have 

(Be, - 36, da'lx, yy D = tr, 

which expresses that the evectant of the discriminant is equal to the quadric ; 

(a, 6, cja^, -a^)« 1/^=40, 
which expresses that the provectant of the quadric is equal to the discriminant ; 

(a, 6, c^bx + cy, —ax — hyf = D tT, 

which expresses that a transmutant of the quadric is equal to the product of the 
quadric and the discriminant. 

94. When the quadric is expressed in terms of the roots, we have 

a-^ U = (x-ay)(X'-fiy\ 

and in the case of a pair of equal roots, 

a''U^(x-OLyy, 
D =0. 

95. The problem of the solution of a quadratic equation is that of finding a 
linear factor of the quadric. To obtain such linear factor in a symmetrical form, it 
is necessary to introduce arbitrary quantities which do not really enter into the solution, 
and the form obtained is thus in some sort more complicated than in the like 
problem for a cubic or a quartic. The solution depends on the linear transformation 
of the quadric, viz. if we write 

(a, 6, c$\x + fiy, vx + pyy = (a', h\ c'Ja?, y)*, 

80 that 

a'— (a, 6, c$X, v)\ 

V = (a, 6, c$X, i/J/A, p), 

c' = (a, 6, c\fi, p)», 

then 

a'c - 6'« = {ac - ¥) (\p - fipy, 

an equation which in a different notation is 

(a, 6, cja:, y)«.(a, 6, c$Z, Yy-{(a, h c^x, yJZ, F))^ = D (F*-Xy)«. 



156] 



A FIFTH MEMOIR UPON QUANTICS. 



529 



in which form it is a theorem relating to the quadne and its first and second 
emanant& The equation shows that 

(a, h, clx, ylX, F) + V rn ( Kr - Zy), 

where (X, Y) are treated as supernumerary arbitrary constants, is a linear factor of 
(a, 6, c\xy yy, and this is the required solution. 

96. In the case of two quadrics, the expressions considered are 

(a, 6, c'^x, y)\ (1) 

(a', h\ c'\x, y)\ (2) 

ac-h" , (3) 

ac'-266'4-ca', (4) 

6" , (5) 



ac 




> » 



(6) 



(o6' - a'b, 
(Xa + fia', 



ac —ac 



be' - Vc \x. yf, 
\c+nc' ^x, yy, 



{ac -b* , ac'- 266' + ca', a'c' - 6'» \\, ft)', 



(7) 
(8) 
(9) 



(1) and (2) are the quadrics, (3) and (5) are the discriminants, and (4) is the lineo- 
linear invariant, or connective of the discriminants ; (6) is the resultant of the two 
quadrics, (7) is the Jacobian, (8) is an intermediate, and (9) is the discriminant of the 
intermediate. And where it is convenient to do so, I write 

(1) = u, 

(2) = v. 

(3) = D, 

(*) = Q. 

(5) = □'. 

(6) = R. 

(7) = H. 

(8) = W. 

(9) = e. 



C. II. 






530 



A FIFTH MEMOm UPON QUANTIC8. 



97. The Jacobian (7) may also be written in the form 

a, b , c 

The Resultant (6) may be written in the form 



[156 



a, 


26. 


a, 26, 


c. 


a\ 


26'. 


a', 26', 


c'. 



and also, taken negatively, in the form 

4 {ab' - a'6) (6c' - Vc) - {ac' - a'c)», 
which is the discriminant of the Jacobian ; and in the form 

4 (oc - ¥) (a V - 6'») - {ac' - 2hV + caj, 
which is the discriminant of the Intermediate. 

98. We have the following relations: 

(a, 6, c\Vx + c'y, - a'x - 6'y)» = - (aV - 6'*) (a , 6 , c\x, yf 

+ (ac'-266'4-ca') (a', 5', c'$a?, y)«, 

(a', 6', c' \hx -\-cy,-ax- byY = + (oc' - 266' + ca') (a , 6 , c $a?, y)» 

-(ac-6«) (a', 6', c'l^x, y)\ 
and moreover 

(ac-6«, ac' -266' 4- ca', a'c'-6'«$£r, - tO* 

= - {{ab' - a'6, oc' - a'c, 6c' - 6'c][a', y)*}*, 

an equation, the interpretation of which will be considered in the sequel. 

"99. The most important relations which may exist between the two quadrics are 
First, when the connective vanishes, or 

ac' - 266' + ca' = 0, 



in which case the two quadrics are said to be karmonically related: the nature of 
this relation will be further considered. 



156] A FIFTH MEMOIR UPON QU ANTICS. 531 

Secondly, when i2 = 0, the two quadrics have in this case a common root, which 
is given by any of the equations, 

= 3ii'B : dff'R : d^R 

= 6c' — Vc : ca' — c'a : aV — a'6. 

The last set of values express that the Jacobian is a perfect square, and that 
the two roots are each equal to the common root of the two quadrics. 

The preceding values of the ratios a^ : 2xy : y' are consistent with each other in 
virtue of the assumed relation i2 = 0, hence in general the functions 

4SaR . dcR - (djty, daR . db'R - dbR . daR, &c. 

all of them contain the Resultant i2 as a factor. 

It is easy to see that the Jacobian is harmonically related to each of the quadrics; 
in fact we have identically 

a(6c'-6'c)4-6 (ca'-c'a)4-c (a6'-a'6) = 0, 
a' (6c' - 6'c) + V {ca' - c'a) 4- c' (a6' - a'6) = 0, 

which contain the theorem in question. 

100. When the quadrics are expressed in terms of the roots, we have 

a-^U =(a?-ay)(a?-^y), 

a'-'W =(^-a'y)(j:-^'y), 

4a-^n =-(a-)8)«, 

2 (aa')-' = 2a^ + 20^)8' -{a -^13) (a' + /S'), 

4a'-^n' =-(a'-^')», 

(aa^-^R =(a-a')(a-^')()8-a')(/S-/8'), 



(oaT'H = 



y«, 2yx , x^ 
1, a+^, afi 
1, a' + iS', o^ff 



101. The comparison of the last-mentioned value of -K with the expression in 
terms of the roots obtained from the equation 

-ij=4nn'-Q», 

gives the identical equation 

which may be easily verified. 

67—2 



532 



A FIFTH MEMOIR UPON QUANTIC8. 



[156 



102. We have identically 

2a/8 + 2a'/8' - (a + yS) (o' + /S') 

= 2 (a -«')(« -/SO -(a -/9)(2a 
= 2(/3-a')(y8 -^)-(fi -a)(2ff 
= 2(0' -a)(a' - /3 ) - (a' - /T) (2a' 
= 2 09' - a ) (/y - yS ) - (/3' - 0') (2/3' 



a: 

a' 
a 

a 



/9); 



and the equation Q^^ac' — 266' + ca' = . njay consequently be written in the several 
forms 



a-/S a-a' a-fi 
2 1 1 



/ > 



/3-a a-a' ' fi'-fi' 



1 1 

+ -r 



a'-/3' a'-a ' o'-/8' 



1 1 

+ 



so that the roots (a, yS), (a', /3') are harmonically related to each other, and hence the 
notion of the harmonic relation of the two quadrics. 

103. In the case where the two quadrics have a common root a = a', 



a-' 


U 


= (^- 


ay) {x - 


-/8y), 


a'-' 


U' 


= (x- 


ay) {x - 


-/9'y). 


4a-*n 


= -(a 


-/8)'. 




2 (aa')-'Q 


= («- 


|8)(a- 


■n 


4a'- 


-,□' 


= -(a 


-yS')'. 




R 




= 0, 







(m')-' H = (^ - 0) (X - a^y. 

104. In the case of three quadrics, of the expressions which are or might be 
considered, it will be sufficient to mention 



(a , b , c Ijix, yY, 
(a ,h',c' $ar, yf. 
(a", b", c"\x, yy. 

a , b , c ' > 
a , , c 
a", b", c" 



(1) 
(2) 
(3) 

(4) 



156] 



A FIFTH MEMOIR UPON QU ANTICS. 



533 



where (1), (2), (3) are the quadrics themselves, and (4) is an invariant, linear in the 
coefficients of each quadric. And where it is convenient to do so, I write 

(1) = u, 

(2) = W, 

(3) = U". 

(4) = n. 

105. The equation H = is, it is clear, the condition to be satisfied by the 
coefficients of the three quadrics, in order that there may be a syzygetic relation 
\U + fjkU' + vU" = 0, or what is the same thing, in order that each quadric may be 
an intermediate of the other two quadrics; or again, in order that the three quadrics 
may be tn Involution. Expressed in terms of the roots, the relation is 



1, a +13, afi 
1, af+^, a[ff 
1, a" + ^", a";8 



n Qii 



= 0; 



and when this equation is satisfied, the three pairs, or as it is usually expressed, the 
six quantities % fi\ a!, ff \ ol\ ff\ are said to be in involution, or to form an 
involution. And the two perfectly arbitrary pairs a, )8; oi^ ff considered as belonging 
to such a system, may be .spoken of as an involution. If the two terms of a pair 
are equal, e.g. if a" = ff' = ^, then the relation is 



1, 2^ , ^ 
1, a+)8, dfi 
1, o' + ^S', alff 



= 0; 



and such a system is sometimes spoken of as an involution of five terms. Con- 
sidering the pairs (a, fi\ (a', )8') as given, there are of course two values of 6 which 
satisfy the preceding equation; and calling these 6^ and 6^^, then 6, and 0,^ are said 
to be the sibiconjugates of the involution a, ^8; a', ff. It is easy to see that ^^, 6^^ 
are the roots of the equation H=0, where H is the Jacobian of the two quadrics 
U and IT whose roots are (ot, /8), («', ff). In fact, the quadric whose roots are 0,y 0„ is 

y\ 2yx , od" 
1, a+)8, ap 
1, a' + ^, a'^ 

which has been shown to be the Jacobian in question. But this may be made clearer 
as follows: — If we imagine that X, fi are determined in such manner that the inter- 
mediate \V -¥ fiir may be a perfect square, then we shall have \U'{-fiir — a''{x'-0yY, 
where denotes one or other of the sibiconjugates 0^, 0^^ of the involution. But the 
condition in order that \U •{■ filT may be a square is 

{ac - h\ ad - 266' + ca, dd - y'\\ fiy ; 



534 



A FIFTH MEMOIR UPON QUANTICS. 



[156 



and observing that the equation \ : fi= W : -^-U implies \U+ filT ^0=^a'' (x— OyY, it 
is obvious that the function 

must be to a fetctor prh equal to (x — O^yf (x — 0„yy. But we have identically 

(ac - 6«, ac' - 266' + ca\ ale' - V'^W, ^U)'^^ {{ab' - a% ac' - a'c, be' - Vcl^x, y)»}«, 
and we thus see that (x — 0^y), (x — 0^^y) are the £Eu;tors of the Jacobian. 

106. It has been already remarked that the Jacobian is harmonically related to 
each of the quadrics U, U'; hence we see that the sibiconjugates 5,, 0„ of the 
involution a, y3, a^, ff are a pair harmonically related to the pair a, /3, and also 
harmonically related to the pair a\ ^, and this properly might be taken as the 
definition for the sibiconjugates 5,, 0,^ of an involution of four terms. And moreover, 
a, P\ a\ fi' being given, and 0^, 0^^ being determined as the sibiconjugates of the 
involution, if a", /S^' be a pair harmonically related to 0,, 5,,, then the three pairs 
a, /9; a', )8'; a", /8" will form an involution; or what is the same thing, any three 
pairs a, /8; oi, ^\ a", ^", each of them harmonically related to a pair 0,, 6^^, will be 
an involution, and 0^ , 0^^ will be the sibiconjugates of the involution. 

107, In particular, if a, yS be harmonically related to 0^, 5,^, then it is easy to 
see that ^^, 0^ may be considered as harmonically related to d^, d^^, and in like manner 
^//» ^// wi^' ^^ harmonically related to 5,, 0^/, that is, the pairs 5,, 0/^ 0^^, 0^^ and 
a, fi will form an involution. This comes to saying that the equation 

1, ^e, . e; =0 

1, a + /3, «^ 



is equivalent to the harmonic relation of the pairs a, /S; B^, 0^/, and in &ct the deter- 
minant is 

(d, - 0„) {2afi + 20,0,, -(a + 13) (0, + 0,,)), 

which proves the theorem in question. 

108. Before proceeding further, it is proper to consider the equation 

= 0, 



1, a, 


a', 


aal 


1. fi. 


^, 


^ff 


1. 7. 


i. 


r/ 


1, 8. 


8'. 


^ 



which expresses that the sets (a, yS, 7, S) and (a', ff, y\ S^ are homographic; for 
although the homographic equation may be considered as belonging to the theory of 



156] A FIFTH MEMOIR UPON QU ANTICS. 535 

the bipartite quadrie {x — ay) (x — ay), yet the theory of involution cannot be completely 
discussed except in connexion with that of homography. If we write 

^ =(/S-7)(«-8), 5=(7-a)(^-8), C ={a - p)iy -i), 
A' = (^-Y)(c^-n B' = W-c^)(B'-n (7 = (a' - jSO (7' - 8'), 
then we have 

and thence 

BC-FC^CA''-C'A^AR^A'B; 

and either of these expressions is in fact equal to the last-mentioned determinant, as 
may be easily verified. Hence, when the determinant vanishes, we have 

A : B : C^A' : R : C, 
Any one of the three ratios A : B : C, for instance the ratio B : C,= 

(7-«)(/8"g) 
(a-)8)(7-8)' . 

is said to be the anharmonic ratio of the set (a, ^, 7, S), and consequently the two 
sets (a, ^, 7, S) and (a\ ff, 7', S') will be homographically related when the anharmonic 
ratios (that is, the corresponding anharmonic ratios) of the two sets are equal. 

If any one of the anharmonic ratios be equal to unity, then the four terms of 
the set taken in a proper manner in pairs, will be harmonics; thus the etiuation 

^ = 1 gives 

which is reducible to 

2aS 4. 2^7 - (a + S) 08 + 7) = 0, 

which expresses that the pairs a, B and 13, 7 are harmonics. 

109. Now returning to the theory of involution (and for greater convenience 

taking a, 0^ &c. instead of a, )8 &c. to represent the terms of the same pair), the 

pairs a, a'; )8, iS'; 7, 7'; S, 8'; &c. will be in involution if each of the determinants 
formed Mrith any three lines of the matrix 

1, a +a! , aa' , 
1, fi + fi', pff, 

^ 7+7» 77'» 

1, S+S', SS', 
&c. 



536 



A FIFTH MEMOIR UPON QUANTICS. 



[156 



vanishes: but this being so, the determinant 



which is equal to 



1, 


a, a', 


aa' 


1. 


A ^. 


18/8' 


1, 


7. 7'. 


77' 


1, 


8, 8', 


SS' 


«■ 


1, « + «' 


, aa' 


/9. 


1, 13 + ^ 


> fi^ 


7. 


1. 7+7' 


. 77' 


B, 


1, B +B' 


, SS' 



will vanish, or the two sets (a, )8, 7, S) and (a', /S', 7', SO will be homographic ; that 
is, if any number of pairs are in involution, then, considering four pairs and selecting 
in any manner a term out of each pair, these four terms and the other terms of 
the same four pairs form respectively two sets, and the two sets so obtained will be 
homographic. 

110. In particular, if we have only three pairs a, a'; fi, ff\ 7, 7', then the sets 
a, )8, 7, a' and a', ff, 7', a will be homographic; in fact, the condition of homography is 



which may be written 



or what is the same thing, 



1, 


a, 


a', aa! 


= 


} 


1, 


A 


^. P^ 




1. 


7. 


7» Ti 




1, 


a'. 


a, ojoi 




a. 


1, 


a 4- a', aa' 


J= 


/9. 


1, 


/S + i8', P^ 




7. 


1. 


7 + 7'. 77' 




a'. 


1. 


a +0', ew' 




a 


> 


1, a + a* , oa' 


/3 


9 


1. iS + yS', y9/8' 


7 


1 


1. 7+7'. 77' 


0'- 


-a, 


0, 


. c 


) 



= 0, 



so that the first-mentioned relation is equivalent to 



(«' - «) 



1, a + a' , ao* | = 0, 
1, y8 + ^, /SyS- 

1. 7+7'. 77' 



156] 



A FIFTH MEMOIR UPON QUANTICS. 



537 



and the two sets give rise to an involution. The condition of homography as expressed 
by the equality of the anharmonic ratios may be written 



a-7.a'-^'"a'-7'.a-/8'' 



or multiplying out, 



(a - ;8) (a - ;80 («' - 7) (0' - yO - («' - ;8) (a' - )8') (a - 7) (a' - 7') = 0. 

which is a form for the equation of involution of the three pairs. But this and the 
other transformations of the equation of involution is best obtained by a different 
method, as will be presently seen. 

Ill, Imagine now any number of pairs a, a'; fi, /3^; 7, 7'; S, S'; &c. in involution, 
and let x, y, z, w he the fourth harmonics of the same quantity X with respect to 
the pairs a, a' ; fi, 13" ; 7, 7' and B, 8' respectively ; then the anharmonic ratios of the 
set (Xy y, z, w) will be independent of X, or what is the same thing, if x\ y\ /, w 
are the fourth harmonics of any other quantity X' with respect to the same four pairs, 
the sets (a?, y, ^, w) and (x\ y', /, w') will be homographic, or we shall have 

= 0. 



1, 


X, 


x\ 


OCX 


1, 


y> 


y'. 


yy 


1, 


z. 


«', 


Z!f 


1, 


w, 


w', 


vm 



It will be sufficient to show this in the ceese where X is anything whatever, but X' 
has a determinate value, say X' = 00 ; and since if all the terms a, a', &c. are 
diminished by the same quantity X the relations of involution and homography will 
not be affected, we may without loss of generality assume X = 0, but in this case 



X = — — , , a;' = i (a -h a'), 



and the equation to be proved is 



1. 



aa 



1, 



1, 



1, 



a + a" 
77' 



/ f 



7+7 

SB' 
B + B^' 



a -f flf, aa* ; = 0, 



/3 + /8', 13^ 



7+7'» . 77' 



B + B\ BB' 



which is obviously a consequence of the equations which express the involution of the 
four pairs. 

C. II. 68 



538 



A FIFTH MEMOIR UPON QUANTICS. 



[156 



A set homographic with x^ y, z w, which are the fourth harmonics of any quantity 
whatever X with respect to the pairs in involution, a, a'; fi, 13'; 7, 7'; S, B\ is said to 
be homographic with the four pairs, and we have thus the notion of a set of single 
(quantities homographic with a set of pairs in involution. This very important theon- 
is due to M. Chasles. 

112. Let r; 8] t he the anharmonic ratios of a set a, fi, 7, B, and let r/, s,; t, 
be the anharmonic ratios (corresponding or not corresponding) of a set a^, ^^, 7^, h^ And 
suppose that /; ^; t'; r/; <; e/; r"; *"; T; <; <'; C; ^'"; «'"; ^"; r/"; <"; C 
are the analogous quantities for three other pairs of sets ; then an equation such as 



TV, TV, 



= 0. 



or as it is more conveniently written, 



»«, > 


»•». . 


»•/« . 


»^, 


«v , 






rv; 


A" . 


»^V' . 


r/V . 




«"V", 


r'V. 


r/'V", 


r r, 



= 



is a relation independent of the particular ratios r : 8 which have been chosen for the 
anharmonic ratios of the sets; this is easily shown by means of the equations 

r + « + « = 0, r, + «, + «, = 0, 

which connect the anharmonic ratios. The equation in fact expresses a certain relatiou 
between four sets (a, ^, 7, S) and four other sets (a^, /8^, 7^, S^); a relation which may 
be termed the relation of the homography of the anharmonic ratios of four and four 
sets : the notion of this relation is also due to M. Chasles. 

113. The general relation 

1, a + /8 , a/3 =0 

1, ot +^', f£ff 
1, a" + i8", ot'P' 

may be exhibited in a great variety of forms. In fact, if the determinant is denoted by 
T, then multiplying by this determinant the two sides of the identical equation 



w', — M, 1 
v", -V, 1 

V^^ "W, 1 

we obtain 

T (u — v) (v — w) (w — w) = 



= (w — v) (v — w) (w — u\ 



(M-a)(t/-/3), (t;-a)(t;-)8), (w-a )(«;- )8 ) !. 
(^-a')(u-/9^), (t;-a')(t;-/}'), (w^a')(w^^) 
(^-0(^-/9^), (t^-a'Xt'-rX {wa")(w^l3'') 



156] 



A FIFTH MEMOIR UPON QUANTICS. 



539 



If, for example, u = a, v = yS, then we have 

T (a - ;8) = - (a - a') (a - /90 C9 - O (/9 - /8") + (/3 - «') (/3 - ^) (« - «") (« - /3") ; 
and again, if « = a, v = a', v> = a", then we have 

T = - (a - /3") (a' - ;8) (a" - i80 + (a - /S-) («' - /9") («" - yS). 

Putting T = Q, the two equations give respectively 



(g - a') (ff - c^') _ (« - yyp (/3 - /y) . 

(a-«")(«'-/8)~(«-i8')(i9"-/9)' 



and 



(« - /8") («' - /3) («" - /8') = (a - yS-) («' - r) («" - /8), 
which are both of them well-known forma 

114. A corresponding transformation applies to the equation 



which expresses the homography o 
representing by V the similar determinant 



, a, of, aal =0, 

, A ff. (iff 

> 7» i> Tl 

, &, O y CO 

two pairs. In fact, calling the determinant 'V and 



V9 = 





ss' , — »' , 


-s, 1 


> 








ttf, -if. 


-t, 1 










uu', — «', 


-tt. 1 










m/ , —v'. 


-V. 1 








ted to zero, would express the 
we have 


homography of 


the sets («, t, 


II, v) and 


(«-«)(«'-«'). {8-fi)(8'-n 


(«-7)(«'-7). 


(«-«)(*' -80 


9 


(t - a) (f - a'), {t-fi)(1f -n 


(t - 7) «' - 7 ). 


(<-«)(«'-«') 




(« - a) («' - a!), (u - /S) (u' - ^, 


(« - 7) («' - y). 


(m-8)(m'-S0 




(v-a){t/-af), (v- 


dyw-n 


(» - 7) (» 


-7). 


(» - S) (t/ - 8') 





which gives various forms of the equation of homography. In particular, if « = o^ s' = ff', 
t = ff, (f = a', u = y, «' = S', » = S, v'=y, then 



7^ = 



(« 
(/3 



7)03' 
7) (a' 



70, (« 
70. (-8 



S)08'-80 



(7 - «) (S' 
(8 - a) (y 



«0. (7 



/8)(8' 
y8)(7' 



/SO 



68—2 



540 



A FIFTH MEMOIR UPON QUANTICS. 



[156 



and the right-hand side breaks up into factors, which are equal to each other (whence 
also V = '^), and the equation S?^ = takes the form 

(« - 7) (/8 - S) («' - S') (/S- - 7') - (« - 8) (-8 - 7) («' - 7') (/S' - «') = 0. 

which is, in fact, one of the equations which express the equality of the anharmonic 
ratios of (a, )8, 7, S) and (a', ff, y\ S^. 



115. In the case of a cul:)ic, tlie expressions considered are 

(a, 6, c, d$a?, y)», 
(ac — b^ ad — be, bd — t^^a;, yf, 

- a^d + %abc - 26» '\ 

- abd + 2a(^ - 6«c 
■\-acd- 2b^d 4- 6c* 

[ -\-ad' - Sbcd + 2c* ; 

a«(? - 6abcd + 4ac» 4- 46»df - 36V, 



(1) 
(2) 



[^> y)** 



(3) 



(4) 



where (1) is tlie cubic, (2) is the quadrico variant or Hessian, (3) is the cubico variant, 
and (4) is the quartinvariant or discriminant. 

And where it is convenient to do so, I write 



(1) 

(2) 

(3) 
(4) 



U, 
H. 



so that we have 



*»-ni7»4-4JJ» = 0. 



116. The Hessian may be written under the form 

(cur + by)(cx-\-dy) — (6a? 4- cy)*, 

(which, indeed, is the form imder which qua Hessian it is originally given), and under 
the form 

a, b ., c 
6 , c , (Z 

The cubicovariant may be written under the form 

{2 (ac - 6") a? + (ad - be) y] (6a^ + 2cxy 4 dy») 
- { {ad-bc)x + 2{bd - c")y} (cue* + 2&py 4- cy*), 



156] A FIFTH MEMOIR UPON QU ANTICS. 541 

that is, as the Jacobian of the cubic and Hessian ; and under the form 

that is, as the evectant of the discriminant. 

The discriminant, taken negatively, may be written under the form 

+ 4 (oc - b^)(bd -'(^)-{ad- hcf, 
that is, as the discriminant of the Hessian. 

117. We have 

(a, 6, c, d\ha^ + 2ca>y + df, -aaf- ihayy - c}/J = ir<^, 

which expresses that a transmutant of the cubic is the product of the cubic and the 
cubicovariant. The equation 

{(3a, 9^, 3c, 9dl[», -^)'}«n=2Z7' 

expresses that the second evectant of the discriminant is the square of the cubic. 

The equation 

# . -3cd , -36d + 6c« , -36c-h2ad = 27 D' 

-3cd , -3c« +126d, -3ad-66c , -3ac + 66« 

-36d-f6c» , -3arf-66c , -36« + 12ac, -3aft 

I 

- 36c - 12ad, - 3ac + 66' , 3a6 , a« ' 

expresses that the determinant formed with the second differential coefficients of the 
discriminant gives the square of the discriminant. 

The covariants of the intermediate aU-h /3^ are as follows, viz. 

118. For the Hessian, we have 

H(aU-^l3<P)= (1, 0, -n3[a, /3yH 

= (a>-/8»n)ir; 
for the cubicovariant, 

^(a[7+^4>)= (0, D, 0, -n» 5a, fifU 

+ (1, 0, -D, 05a,^)»c|> 

and for the discriminant, 

Q(a£r + ^4>)= (1, 0, -2D, 0, D^^a, fi)*^ 

= (a«-/S«n)«D, 

where on the left-hand sides I have, for greater distinctness, written JV, &c. to denote 
the functional operation of taking the Hessian, &c. of the operand aU + fi^, 



542 A FIFTH MEMOIR UPON QUANTICa [156 

In particular, if a = 0, ^ = 1, 

119. Solution of a cubic equation. 

The question is to find a linear factor of the cubic 

(a, 6, c, d$a?, yY, 
and this can be at once effected by means of the relation 

between the covariants. The equation in fact shows that each of the expressions 

is a perfect cube, and consequently that the cube root of each of these expressions 
is a linear function of (x, y). The expression 

is consequently a linear function of x, y, and it vanishes when [7 = 0, that is, the 
expression is a linear factor of the cubic. 

It may be noticed here that the cubic being a(a? — ay)(a? — ^y)(a? — 7y), then we 
may write 

^Ji^Tu^) - ^^(^-U^U) = J a(a) -tti>)(^ -7)(a?-ay), 

where 6> is an imaginary cube root of unity: this will appear firom the expressions 
which will be presently given for the covariants in terms of the roots. 

120. Canonical form of the cubic. 

The expressions i(4>+ [/VD), i(4>— tTVO) are perfect cubes; and if we write 

then we have 

U^ x» + y», 

4) = Vn (x» - y»), 
and thence also 

ir=-^nxy. 



156] 



A FIFTH MEMOIR UPON QUANTICS. 



543 



121. When the cubic is expressed in terms of the roots, we have 

a-^fT^ (x - ay){x - fiy)(x - 7^) ; 

and then putting for shortness 

^=0-ry)(a:-ay), B ^ {r^ -■ a) {x -- fiy\ C = (a - ^){x -yy\ 

so that 

Jl+£ + C = 0, 

we have 

a-*^ =^^{B''C){C^A){A^B\ 

122. The covariants -ff, 4> are most simply expressed as above, but it may be 
proper to add the equations 



a" 



'a» + /8* + 7* - ^7 - 7a - aA 
= -ij 6a^7-i87«-7a»-a)3»-)8V-7«a-a«/3, jTo?, y^ 

= - 4 {(a + ©i8 + ai«7) a? + (^7 + tt)7a + w'^aiS) y 1 {(« + ©'^/S -h 0)7) a- 4- (^7 4- ©V* + ®«i8) 3^] 



(where © is an imaginary cube root of unity), 

a-^4>=^2(a-^)(a-7)«(a:-^y)«(ar-7y) 

' 2(a« + ^' + 7')-3(^ + 7a' + aiS" + )8»7 + 7*a + o?/3) + 12a^7, 

-2(a«/87 + /8Va + 7'a^) + 4(^' + 7»a« + a»^)-(i87» + 7a« + a^ + ^7 + y^ 

- 2 (a^V+)87'a*+7aW+4(a»^7+i8V+7*«i8)-(^+7»aHa«/9'+^^ ' "^ 

^+2(^V + 7V + a»/3»)-3(a)8V + )87V + 7a«/3» + a^V + ^7'«' + 7<3^i8") + 12^^ ; 

= {(2a-^~7)a;+(2^7-7a~a^)^} K2^-7-a)^+(27a -,7^-^7)3^} {(27-a-^);r+(2a^-/87-7a)yj 

123. It may be observed that we have a-^UU^ = - ^ A^B^O, which, with the 
above values of H, 4> in terms of A, B, C and the equation A + B-hG = 0, verifies 
the equation 4>' — OU* + 4iH^ = 0, which connects the covariants. In fact, we have 
identically, 

{B^cyiC-AyiA-By^^ 

-'4(A-^B + CyABC-^{A+B-hC)^{BC-^CA + ABy-^lS{A+B-^C)(BC-{-CA-{-AB)ABa 

-^(BC+CA+ABy- 27 A'B'O, 

by means of which the verification can be at once eflfected. 



544 A FIFTH MEMOm UPON QUANTICS. [156 

124. If, as before, cu is au imaginary cube root of unity, then we may write 

27a-»4> =-(£-C)(C-A)(il~5), 

27a-» fT Vn = 3 (ft) - ft)«) ABC, 
and these values give 

27a-»i(4)+ U'JD = [{a + 0)^ + arf) X -^ (By + a)^a + <oa^ ) y}\ 

27a-» J (^ - ^ ^3 = ((« + <»^ + ^'t) a? + (^87 + «7a + «'a/8) y}'» 
and we thence obtain 

^J(4>+£rVn) - ^i(4>- [TVd) = _ Ja (ft> - fti')(i8 - 7)(ar - ay), 
which agrees with a former result. 

125. The preceding formulae show without difficulty, that each factor of the cubi- 
covariant is the harmonic of a factor of the cubic with respect to the other two factors 
of the cubic; and moreover, that the factors of the cubic and the cubicovariant form 
together an involution having for sibiconjugates the factors of the Hessian. In feet, the 
harmonic of x — ay with respect to (^- /8y)(^ — 7y) is (2a — )8- 7)0? + (2^7 — 7a — a/8)y, 
which is a factor of the cubicovariant ; the product of the pair of harmonic factors is 

(2a - ^ - 7)a:« + 2 087 - a^)xy + (- 2a^7 + a^/S + a^) yM 

and multiplying this by fi — y, and taking the sum of the analogous expressions, this 
sum vanishes, or the three pairs form an involution. That the Hessian gives the sibi- 
conjugates of the involution is most readily shown as follows: — the last-mentioned 
quadric may be written 

(-(a + ^ + 7) + 3a)a;» + 2(a^4-a74'i87-a(a + ^ + 7>)a:y + (--3a^7 + a(a)8 + a7 + /87))y*» 
which is equal to 

or, throwing out the factor 3a~S to 

(6 + oa, 2c - 26a, d -f ca^x, y)\ 

which is harmonically related to the Hessian 

(oc — 6^, ad — be, bd^-c^^x, yY\ 

and in like manner the other two pairs of factors will be also harmonically related to 
the Hessian. 



156] 



A FIFTH MEHOm UPON QUANTICS. 



545 



126. In the case of a pair of equal roots, we have 



a-'U= 


(x - ayf {x - yy) , 


a-^H = 


- 4 i<i-yyix-2yy, 


a-^ = 


- lA- (« - 7)* (« - ay)*. 


D = 


0. 



And in the case of all the roots equal, we have 

H = 0, 4) = 0, n=o. 

127. In the solution of a biquadratic equation we have to consider the cubic 
equation tj' - Jlf (tj - 1) = 0. The cubic here is (1, 0, —if, M\w, ly, or what is the 
same thing, 

(1. 0, - ii/, if $«r, 1)»; 
the Hessian is 

M(-i. 1, -iM'^w, 1)»; 
the cubicovariant is 

if(-l, filf, -p/, if+^ilf>$t!r, 1)»; 
and the discriminant is 

if«(l-^if). 



128. 



In the case of a quartic, the expressions considered are 

(a, 6, c, d, e^x, yY, 

ac - iibd -f Sc*, 
(ac - 6», 2 (ad- 6c), ae-\-2hd- Sc', 2 (6e - cd), ce- d?\x, y)\ 

ace + 2bcd — ck? — J*6 — c*, 

^- a«d+ 3a6c- 2&», "j 

- a»c - 2 abd-\- 9 oc* - 6 6»c, 

- 5abe-^ 15 acd - 10 b^d, 
f 10 ad« - 10 6«g, 
+ 5 ade+ 10 6d* - 15 bee, 
+ ae» + 2bde- 9(fe -\-Gcd\ 
+ 6e« - 3 c(fo + 2 d^* 



^a^, y)*, 



(1) 

(2) 

(3) 
(4) 



(5) 



where (1) is the quartic, (2) is the quadrinvariant, (3) is the quadricovariant or Hessian, 
(4) is the cubinvariant, and (5) is the cubicovariant. 



C. II. 



69 



546 A FIFTH MEMOIR UPON QUANTICS. [l56 

And where it is convenient to do so, I write 

(1) = u. 

(2) = / . 

(3) = H. 

(4) = /, 

(5) = <t>. 

The preceding covariants are connected by the equation 

The discriminant is not an irreducible invariant, its value is 

D = /' - 27/» = a»e» + &c., 
for which see Table No. 12, [p. 272]. 

129. It is for some purposes convenient to arrange the expanded expression of the 
discriminant in powers of the middle coeflBcient c. We thus have 

D = aV - 12 d'bde^ - 27 a«d* - 6 ab^cfe - 27 6V - 64 6»d' 
+ c ( 54 a*(?« + 54 a6V + 108 abd^ + 108 b'de) 
+ c« (- 18 aV - 180 abde + 36 6»d«) 
+ c» (- 54 Ok? - 54 b'e) 
+ c* (81 ae). 

130. Solution of a biquadratic equation. 

We have to find a linear factor of the quartic 

(a, b, c, d, e$x, yy. 
The equation JU^ — lU^H -\- AiH^^ — ^^, putting for shortness 

may be written 

(1, 0, -M, M\IH, juy = -\p^\ 

Hence, if w,, cr,, ts-j are the roots of 

(1, 0, -if, Afl^tsr, 1)» = 0, 
the expressions IH —miJU, IH — vr^JU, IH—vr^JU are each of them squares; write 

(tjT, - tsr,) {IH - rff.JU) = X\ 
(tsr, - tsrO {IH - tsr, J(/) = 7«, 
(tsy, - tsr,) (/^r - tsr, Ji/) = Z^ , 



156] A FIFTH MEMOIR UPON QU ANTICS. 547 

so that, identically, 

and consequently X -i-iY, X — lY are each of them squares. The expression 

aX -\- I3Y + yZ 
will be a square if only 

sua may be seen by writing it under the form 

and in particular, writing Vtjj — btj, Vbt, — btj, Vbti — ta-j for a, )8, 7, the expression 

is a square; and since the product of the different values is a multiple of U^ (this 
is most readily perceived by observing that the expression vanishes for U = 0), the 
expression is the square of a linear factor of the quartic. 

131. To complete the solution: tjj, Wa, tj, are the roots of the cubic equation 

(1, 0, -iif, Jl/Jtsr, 1)» = 0; 
and hence, putting for shortness, 
P» = iif {(-1, fif, -JJlf, if+^if»$/^, Jf^)» + Vl-,Vif(l, 0, -JJlf, ilfj/if, JUf, 
<?=iif{(-l, |ilf, -lM,M+^iP^IH, JUy - s/T^^M {\, 0, -Jif, 3/$//f, JfT)', 

we have (o) being an imaginary cube root of unity) 

i (® - «') («^« - «^3 ) (/^ - «^i .^ fO = ^ - ; 
and if 



Q,» = iif {-l-Vl-^i/}, 
then 

J (w - ft)«) (iSTa - tsr,) = Po - Oo- 

Hence, multiplying and observing that (o) — cd')' = — 3, we find 

- (^h^f ^"^ - ^'^^ ^^^ -^xJU)=(p- 0) (P. - 0.). 

and consequently 

(w, - w,) 'JiH-vfJU = (o) - 6)») V - (P - Q) (P, - Q,). 

We have, in like manner, 

i(«-«')K-«^.)(/^-«r,Ji/)= ^- Q. 

H« - «*) («. - ^i) iIH-w,JU) = a>P- »»Q, 
i (w - «») («, - «,) (/fl^ - nr,JU) = m*P-a> Q, 

69—2 



548 A FIFTH MEMOm UPON QU ANTICS. [156 

and J (© - ai«) (isr, - tsr,) = Po - Oo, 

4 (ft) - CD*) (tSr, - tSTi) = <» ^0 - o^'Qo, 
^ (ft) - O)') (iSTi - -CTj) = ©'Po - O) Qo, 

and therefore 

(m, - t!r,) \/iir^;nT^ {<o - ft)») v- (p-q) Tj^v^Qo), 

(tsr, - tsTj) V/fr "'ST^JU^ia)'' ft)») V - (ft)P - ft)^)(ft)Po - ft)«Q), 
(«r,- tsrO VZH^^^^;jT7 = (ft)- ft)«) V - (ft)«P- cdQ)(^«Pc - ft)Oo) ; 

and hence disregarding the common factor &) — ft)^ the square of the linear factor of 
the quartic is 

V-(P-0)(Po-Oo) + V-(ft)P-ft)«(2)(ft>Po-ft)»(2i,) + V :r(^P-ft)Q)(ft,='Po-ft)Qo), 
which is the required solution. 

It may be proper to add that 

-tJi= Po+ Oo, 

— t3-2 = ft)Po+ft)*Qo, 
-- BTj = ft)'Po 4- ft) Oo. 

132. The solution gives at once the canonical form of the quartic ; in fact, writing 

X + tF= 2 ^(tJa - isr,) (tar, - -bTi) VJ X^ 
Z - tF= 2 \/(t!r,-'cr,)(isr,--tsri) V7y^ 

where X, Y have their former significations, we find, by a simple reduction, 

IE -'srJU^ (tsr, - isr,) •/ (x» + y«)^ 
/if - isr,Jfr= - (isr, - tsr,) /(x" - y'^, 

ig-i^3Jtr=-^^ ^^^^^^" ^V.4xy, 

and thence putting 

g^ tg-, _ ^ (ft) - ft)') (ft)'Po + ft>Oo) 

Wj — tJa (6)'Po — ««>0o) 

we have 

f;'=x* + y* + 6dxy, 

which is the form required. 

133. The Hessian may be written under the form 

(9e, -9d, 9(j, -96, 9al[^, y)'«/, 
that is, as the evectant of the cubinvariant. 



156] A FIFTH MEMOm UPON QU ANTICS. 549 

The cubicovariaut may be obtained by writing the qiiartic under the form 

(aa? + 6y, bx-i-cy, cx-\-dy, dx + ey'^x^ yy, 

and then, treating the linear functions as coefficients, or considering this as a cubic, 
the cubicovariaut of the cubic gives the cubicovariaut of the quartic. 

If we represent the cubicovariaut by 

4) = (a, b, c, d, e, f, g^x, y)\ 
then we have identically, 

ag-9ce + 8d»=0; 
and moreover forming the quadrinavariant of the sextic, we find 

ag- 6bf + 15ce - 10d» = JD, 
where D is the discriminant of the quartic. From these two equations we find 

bf-4ce + 3d«=-^n, 

which is an expression given by Mr Salmon: it is the more remarkable as the left- 
hand side is the quadrin variant of (b, c, d, e, f $«?, y)*, which is not a covariant of the 
quartic. It may be noticed also that we have 

af-3be + 2cd = 0, 
bg - 3cf 4- 2de = 0. 

134. The covariants of the intermediate 

of the quartic and Hessian are as follows, viz. 

The quadrinvariant is 

/ (a£r + 6/8JT)= (/, 18/, 3/»5a, /8)«; 
the cubinvariant is 

J (aU + a^H)^ (J, I\ dIJ, -/'-h54J=3[a, /S)»; 

the Hessian is 

B{aU+6^H)= (1, 0, -3/5a, ^Y H 

+ (0, /, 9/ 5a, ^yU: 
and the cubicovariaut is 

^{aU-¥6^H)== (1, 0, -97, -54J5a, /3)»4>; 
to which may be added the discriminant, which is 

IIl(a£/' + 6/8if) = (l, 0, -187, 1087, 817^ 97277, -29167»l[a, ISyD. 



550 



A FIFTH MEMOm UPON QU ANTICS. 



[156 



135. The expression for the lambdaic is 



6 , c + X, d 
c — 2\, d , e 



= J + X/-4X=». 



If the determinant is represented by A, that is if 

then if Xi, Xj, \i are the roots of the equation A=0, and if the values of 3aA, &c 
obtained by writing X, in the place of X are represented by 9a Aj, &c., then if x, y 
satisfy the equation 

(a, 6, c, d, e^x, y)* = 0, 
we have identically (X, Y being arbitrary), 

Xy-Yx 

+ V-(9e, -ad, Be, -a^TaapTTyA; 

a theorem due to Aronhold. I have quoted this theorem in its original form as au 
application of the lambdaic, but I remark that 

-(a., -Sd, 3c. -9&, da\X, 7yA = -X(a,...$Z, F )* - (ac - 6«, . . .^Z, Yy = --\U'^H' 

if J7', if' are what U, H become, substituting for (a?, y) the new facients {X, Y). More- 
over, we have 



X = - 



/ ' 



for substituting this value in the equation A = 0, we obtain the before-mentioned equa- 
tion «j'— if (cr~l) = 0. We have, therefore, 

-(?e. -3d, 9<., -3», ^a\X, Yy K^^Xr -H- = -j{IH' -J^V), 

and the equation becomes 



( a, 6, c. d. ^.X F$g^y)^/37^ •Jm-Jm,U''+ '^IH'-Jwjr + 'JTU^J^.V 
Xy — IX 



156] 



A FIFTH MEMOIR UPON QUANTICS. 



551 



Moreover, if {x-ay) be a factor of the quartic, then replacing in the formula y by the 
value ax, (x, y) will disappear altogether; and then changing (X, Y) into (a?, y) where 
X, y are now arbitrary, we have 

(«, h, c. d, e-^x, yf {a, 1 ) ^-j^ ^ih-^,JU + ^TR—^JU + ^llT^.W, 

x — ay ' ^ ' 

which is a form connected with the results in Nos. 130 and 131. 



136. We have 



y*, - 



ixy*. 


6a?!/', 


— 4a?i/, a? 


= 6IH-dJU; 


a , 


36 . 


3c , A 




36 . 


3c , 


d , 




6 , 


3c , 


M , e 




3c , 


3d , 


e , 





a, 



6. 

it will appear from the formute relating to the roots of the quartic, that the ex- 
pression 6IH — 9JU vanishes identically when there are two pairs of equal roots, or 
what is the same thing, when the quartic is a perfect square. The conditions in order 
that the expression may vanish are obviously 

6(ac-b^) : 3(ad-6c) : ac + 26d-3c» : 3(6c-cd) : 6(ce-d«) : 9./ 

= r/ : i : C : d : e : I, 



conditions which imply that the several determinants 

6(ac-6»), 3(ad-6c), ae + 26d-3c», 3 (6c -erf), 6(cc-d0 
a , 6 , c , d , c 



all of them vanish, 
determinants are 



we have identically 



If for a moment we write 6H = (a\ h\ c\ d\ c'Ja, y)*, then the 

j a', 6', c', d\ e i 
a, 6, c, rf, e 

ad' — a'd = 3 {he — 6 c), 
eV - c'6 = 3 idc' - d'c). 
a^' - a'c = 3 {hd' - h'd\ 

and the ten determinants thus reduce themselves to seven determinants only, these 
in fact being, to mere numerical factors priSy the coefficients of the cubicovariant ; 
this perfectly agrees with a subsequent result, viz. that the cubicovariant vanishas 
identically when the quartic is a perfect square; 



552 



A FIFTH MEMOm UPON QUANTIC8. 



[156 



137. It may be remarked that the equation 6IH — 9JU = will be satisfied 
identically if 



a = 



e = 



c — <f>* c — <f) 



where (f> !« arbitrary; the quartic is in this case the square of 



( 



Vc^' '^'-'^' v7^^*' ^^- 



If with the conditions in question we combine the equation 7 — (which in this cjise 
ini[ilioh also .7 = 0), we obtain ^ = 0, and consequently 

a __ i _ c _d 

or the (juartic will be a complete fourth power. 

It in easy to express in terms of the coeflBcients a, b\ c\ d\ e of 6H the dilicrent 

dnterniiimnts 

(i, 6, c, rf , 

6, c, d, 6 ; 



wo bav(^ in fa(*t 



ae-bd = ^(c+j- VaV + 46'd' - 3cA , 
8(6d- c»)= i (c' - ^g V^vT4FJ-'37^) , 



^ 



oc - 6' = i a', 

aci — 6c = J 6', 

be —cd=^c\ 

[ ce -(?=:Je', 

wUi'.iu'ii all the above-mentioned determinants will vanish, or the quartic will be a 
fti*.vhtvX fourth power if only the Hessian vanishes identically. 

138. Considering the quartic as expressed in terms of the roots, we have 

a-' U = (x- ay) {x - fiy) (x - yy) (x - By)] 



iiiA if we write for shortness 



which are connected b} 



C = (a-/9)(7-8), 
A+B + C = 0, 



156] 



A FIFTH MEMOIR UPON QUANTIOS. 



553 



then we have 



a-*I = ij(A* + B' + C*)'=-r^{BC + CA+AB), 
a-*J=ji^(B-C)iC-A)iA- B); 



and for the discrimlDant we have 

^^A'B'C, 
and it is easy by means of a preceding formula to verify the equation □ = /' — 27J*. 

139. The formulae show a very remarkable analogy between the covariants of a 
cubic and the invariants of a quartic. In fact 



For the cubic. 
(A=(^-y)(x-tty), 
fi = (y-a)(a;-/3y). 



For the quartic. 

'A = {B-y)ia-B), 
^ B=(y-a)(B-B), 
.C = (a-/3)(7-«); 



and then we have corresponding to each other: 

For the cubic. 
The Hessian, 
The cubicovariant, 
The cubic into the square root of the discriminant. 

140. For the two covariants, we have 



For the quartic. 
The quadrinvariant. 
The cubinvariant. 
The discriminant. 



and 



if for shortness, 






i3 = (S + i8--7-a, 
(B: = (S + ry-a-/3, 



141. We have 



if=V 



Bfi + ya, 8^(7+ a)-7a (S + /8)$a;, y)», 
S7 + a/3, ^(a+/8)-(^(S+7)$a:, y)». 



(B-'Cy(C-'Ay(A--By' 



or putting for shortness 



we have 



^(B--C)(C-A){A^B)' 



M = 



f(^» + £» + (?»)A«; 



c. u. 



70 



554 A FIFTH MEMOIR UPON QUANTICS. [l56 

and it is then easy to deduce 

isri = A(£-a), 

in fact, these values give 

W"! + «'a + t^s =0, 

WltSTj -f- tSTjtJj -f- tJjtSTi ^ — iff, 
'CJjtJjtS'i = ill, 

and they are consequently the roots of the equation tx* ~ if(«r — 1) = 0. 

142. The leading coefficient of IH—miJU is then equal to a* into the following 
expression, viz. 

3^(^« + £»+(?)a-«(ac-6«)-7i^(A« + 5« + C*)(£-C), 

which is equal to 

^(^« + £»+(?){48a->(ac-6»)-4(5-C)}, 

and the term in { } is 

8(fl^ + a7 + aS + ^7 + /8S + 7S)-3(a + ^ + 7 + S)»-4(7-a)(/9-S) + 4(a-/8)(7-8), 

which is equal to 

-3(S + a-^-7)». 

But IH—'oriJU is a square, and it is easy to complete the expression, and we have 
a-^{IH - r!r,JU) = ~ ^frj (A'+ £. + O) j(g+^ « Y - a, - S^+ 7a, S/3 (^ + «) - ^ (S+/9)$a:, y)«}«, 

We have, moveover, 

«r, — «r8 = — 3Ail, 

tT8""'ori = ""3A£, 

and thence 

x(S + a-/8-7, -Sa + /87, «« (iS + 7) - /97 (^ + «)$«, y)* ; 
and taking the sum of the analogous expressions, we find 

0-* {(tsr, - tj,) •^IU-vJU+ («r, - «,) 'JlH-'atJU+ («r, - «•,) '^IH-VfJU] 

which agrees with a former result. 



156] A FIFTH MEMOIR UPON QUANTICS. 555 

143. The equation 7=0 gives 

A : B : (7= 1 : c» : ©', 

where cu is an imaginary cube root of unity; the factors of the quartic may be said 
in this case to be Symmetric Harmonics, 

The equation ^=0 gives one of the three equations, 

A = B, B^C, C==A] 

in this case a pair of factors of the quartic are harmonics with respect to the other 
pair of factors. If we have simultaneously 7=0, «/=0, then 

^=£ = = 0, 

and in this case three of the fiskctors of the quartic are equal 

144. If any two of the linear factors of the quartic are considered as forming, 
with the other two linear factors, an involution, the sibiconjugates of the involution 
make up a quadratic factor of the cubicovariant ; and considering the three pairs of 
sibiconjugates, or what is the same thing, the six linear £Eu;tors of the cubicovariant, 
the factors of a pair are the sibiconjugates of the involution formed by the other two 
pairs of £Eu;tors. 

In fact, the sibiconjugates of the involution formed by the equations 

(x-ay) (x-By) =0, (x- fiy)(x - ryy) = 
are found by means of the Jacobian of these two functions, viz. of the quadrics 

(2, -S~a, 2Sa$a?, y)\ 

(2, -)3-7, 2/37$a:, yY, 
which is 

(S + a-i8-7, -Sa + /3y, Sa(fi + y)^ I3y(8 + a)^x, yY, 

viz. a quadratic factor of the cubicovariant; and forming the other two factors, there is 
no diflBculty in seeing that any one of these is the Jacobian of the other two. 

145. In the case of a pair of equal roots, we have 

a-i [7= (x- ayY {x - yy) (x - By), 
a-^I ^ tV («-7)'(«-«)', 

D = 0, 

a-»4)= A(7-S)»(2a-7-S, yh^-a^ ya^ + Za^-2yaZ\x, yYix-ayy. 

70—2 



556 A FIFTH MEMOIR UPON QU ANTICS. [156 

In the case of two pairs of equal roots, we have 

a-^ U= {x^ ayy (x - 7y)», 

D = 0, 

a-^J?=-^(a-7)«(ic-ay)«(a?-7y)», 

4>= 0; 

these values give also 

6IH--9JU=0. 

146. In the case of three equal roots, we have 

a~* U= (x — ayy (x - Sy), 

7=0, /=o, n=o, 

a-'H=^i^(x'-Sy{2(x^Byy + {x-^ayy}(x^ayy, 

and in the case of four equal roots, we have 

a"* [7= (a?— cry)*, 

7=0, /=o, n=o, 

£r=0, 4) = 0. 

The preceding formulee, for the case of equal roots, agree with the results obtained 
in my memoir on the conditions for the existence of given systems of equalities 
between the roots of an equation. 



Addition, 7th October, 1858. 

Covariant and other Tables (binary quadrics Nos. 25 bis, 29 A, 49 A, and 50 bis). 

Mr Salmon has pointed out to me, that in the Table No. 25 of the simplest 
octinvariant of a binary quintic^ the coefficients — 210, — 17, + 18 and +38 are 
erroneous, and has communicated to me the corrected values, which I have since 
verified : the terms, with the corrected values of the coefficients, are [shown in the Table] 

No. 25 bis. 

[The terms with the erroneous coefficients were alx^cPef, ac^f^, b^d^/*, bc^d^e ; the 
correct values —220, —27, +22, and +74 of the coefficients are given in the Table 
Q, No. 25, p. 288.] 

^ Second Memoir, Philosophical Transactions y t. czlvi. (1S56) p. 125. 



156] A FIFTH MEMOIR UPON QU ANTICS, 557 

Mr Salmon has also performed the laborious calculation of Hermites' 18-thie 
invariant of a binary quintic, and has kindly permitted me to publish the result, which 
is given in the following Table: 

No. 29 a 

[This is the Table W No. 29 a given pp. 299—303, the form being slightly 
altered as appears p. 282.] 

Mr Salmon has also remarked to me, that in the Table No. 50 of the cubin- 
variant of a binary dodecadicS the coefficients are altogether erroneoua There was, in 
fact, a fundamental error in the original calculation; instead of repeating it, I have, 
with a view to the deduction therefrom of the cubinvariant (see Fourth Memoir, 
No. 78), first calculated the dodecadic quadricovariant, the value of which is given in 
the following Table: 

No. 49 a 
[For this Table see p. 319.] 

It is now very easy to obtain the cubinvariant, which is 

No. 50 bis. 

[This is the Table No. 50, p. 319, the original Na 50 with coefficients which 
were altogether erroneous having been omitted.] 

1 Third Memoir, Philosophical Trantactiont, t, oxlvi. (1S56) p. 635. 



558 



[157 



157. 



ON THE TANGENTIAL OF A CUBIC. 



[From the Philosophical Transactions of the Royal Society of London, voL xlviil for the 
year 1858, pp. 461—463. Received February 11,— Read March 18, 1858.] 

In my "Memoir on Curves of the Third Order *'(*), I had occasion to consider a 
derivative which may be termed the "tangential" of a cubic, viz. the tangent at 
the point (w, y, z) of the cubic curve (•$ic, y, -^)' = meets the curve in a point 
(f , 17, f), which is the tangential of the first-mentioned point ; and I showed that when 
the cubic is represented in the canonical form a:^ + y* + z*+ 6lxyz = 0, the coordinates of 
the tangential may be taken to be x(y^ — z^) : y (z^ — a^) : z{a?^y^). The method given for 
obtaining the tangential may be applied to the general form (a, 6, c,/, g^ h, i^j, k, V$jc, y, zy: 
it seems desirable, in reference to the theory of cubic forms, to give the expression of 
the tangential for the general form^; and this is what I propose to do, merely indicating 
the steps of the calculation, which was performed for me by Mr Greedy. 

The cubic foim is 

(a, 6, c, /, g, A, i, j, k, l^x, y, z)\ 

which means 

aa^ + by^ + cz^+Sfy^z+Sgz^x-h^kr^ + Siysi^ + Sjza^ + Shxf + 6lxyz ; 

and the expression for f is obtained from the equation 

^f = (^ / *» c$( j, /, c, i, g, l^x, y, z)\ - (A, 6, i, /, Z, k^x, y, zy)* 
- (a, 6, c, /, flf, A, i, j, k, l^x, y, zy ((Ex + B), 

1 Philosophical TramactioM, vol. cxlvii. (1857), [146]. 

' At the time when the present paper was written, I was not aware of Bir Salmon's theorem (Higher 
Plane Curves, p. 156), that the tangential of a point of the cubic is the intersection of the tangent of the 
cubic with the first or line polar of the point with respect to the Hessian; a theorem, which at the same 
time that it affords the easiest mode of calculation, renders the actual calculation of the coordinates of the 
tangential less important. Added 7th October, 1858. — A. G. 



157] 



ON THE TANGENTIAL OF A CUBIC. 



559 



where the second line is in fact equal to zero, on account of the first factor, which 
vanishes. And GD, Id denote respectively quadric and cubic functions of (y, z\ which 
are to be determined so as to make the right-hand side divisible by a^\ the resulting 
value of f may be modified by the adjunction of the evanescent term 

(aj7 H- hy ^^z) (a, 6, c, /, g, A, i, j, A?, l\x, y, zf, 

where a, h, j are arbitrary coefficients ; but as it is not obvious how these coefficients 
should be determined in order to present the result in the most simple form, I have 
given the result in the form in which it was obtained without the adjunction of any 
such term. 

Write for shortness, 

-R = (i> ff> $y> -a^)* 
'S=(/, i c $y, zy, 

G=(k, I, g Jy, zy, 
^ = (t» /. *» c$y, zy, 

(A, 6, i, /, I, k $fl?, y, zy^{K P, Q \x, 1)», 

(i, /, c, t, 5r, Z $.;, y, zy = {j, 12, S $^. 1)«, 

(a, 6, c, /, flf. A, i, j, Ar, ZJa;, y, zy = (a, B, C, D^x, ly. 

GDiT + B =(GD, B $a?, 1), 

and then for greater convenience writing (A, 2P, Q\x, ly, &c. for (A, P, Q^x, 1)', &c., 
and omitting the {x, ly, &c. and the arrow-heads, or representing the functions simply 
by (A, 2P, Q), &c., we have 

^f = 6 ( j, 212, £f y 

- 3/( j, 212, S )» . (A , 2P, Q) 
+ 3i(j, 212, £f ).(A, 2P, Qy 

- c . (A , 2P, Qy 

- (a, 35, 30, D) . (GD, B ), 



80 that 



which can be developed in terms of the quantities which enter into it. The con- 
ditions, in order that the coefficients of x, of* may vanish, are thus seen to be ' 

DB = bS* - 3fS'Q + SiSQ' - c(?, 
D® - 30B = 6 (612S») - 3/(2£f»P + 412SQ) + Si (212(? + 4SPQ) - c (6PQ»), 

and from these we obtain 



ffi = 



6cA:-3 
/ 6iZ +6 
^ fik +3 

fH -6 


6i^ +6 
c/>fc-6 

r»A: +6 


6c^ +3 
c/Z-6 

t^Z +6 



\y. «)' 



560 



ON THE TANGENTIAL OF A CUBIC. 



[157 



J'c -1 

( 6/t +3 
/* -2 


ftc/-3 
M» +6 
/H-3 


bei +3 
c/'-6 
fP +3 


be' +1 
c/t -3 

t^ +2 



l^y. »)* 



and substituting these values, the right-hand side of the equation divides by a^, and 
throwing out this factor we have the value of f ; and the values of 17, f may be 
thence deduced by a mere interchange of letters. The value for f is 



X* 


^ 


x«z 


xV' 


ar^« 


x^z* 


«/* 


xy^t 




bp +1 


hj'l + 6 


6^"* + 6 


abck + 3 


o^^ — 6 


ahcg — 3 


a6*c + 1 


abcf + 3 , 


c/*' - 1 


ch^k - 6 


chH - 6 


oWZ — 6 


acfk + 6 


acfl + 6 


06/1-3 


a6i« - 6 




/A/- 3 


fhjl -12 


/^V -12 


afH + 6 


a/V+ 6 


q/gri + 3 


a/» + 2 


afH + 3 




hHj + 3 


^•^^ - 6 


jri - 6 


o/TA; - 3 


at^A; - 6 


ai'l - 6 


6cAA; - 3 


6cA/ - 12 






hHl + 6 


^/*^i + 6 


hch^ - 3 


ft^V +24 


6cf + 3 


hhU - 6 


hi^k + 9 






hijk +12 


hijl +12 


hhij + 6 


Hf + 6 


W +12 


hijk +12 


6^At — 6 










6;Z« +12 


cA' - 6 


c/*A; - 6 


6^+8 


hgl^ +24 










chh? - 12 


chkl - 24 


M -24 


cA» - 8 


«;7 +18 










/%• - 6 


/»/ - 6 


/i^A -12 


/W + 6 


c/hk- 6 










fhH - 3 


/i^A/ - 24 


/a;'^ -24 


Pjk -12 


c)fc«Z -24 










/feZ« -12 


fgjk - 24 


/V* - 3 


yAtA; + 3 


rgh+ 6 










fjkl -24 


^^-^ -24 


^/it7 +24 


/W» -24 


/!;7 -18 










AiVfc/ +24 
ij]^ +12 


ghik + 24 

;a^ + 6 

hiP +24 
yH 4 24 


hi^j + 6 
t;7« +12 


t*»; +24 


/^A/ -48 
fhU +12 

yt;^ - 9 
//» -24 
^ +24 
A«A: + 6 
tA:^ +48 







xyz* 


xz^ 


y* 


y3^ 


yV 


y^ 


«^ 




abet — 3 


abc" - 1 


6«y + 3 


6 V + 3 


ftc/!; + 9 


hc^h - 3 


hcg^ + 3 




ac/^ + 6 


ac/i + 3 


hcl^ - 3 


hcfh - 3 


hchi - 9 


6c^Z + 6 


cVA - 3 




a/i^ - 3 


ai» - 2 


b/y - 3 


hckl - 3 


bgU +18 


hcij + 3 


c/i^/ - 6 




6c^A - 9 


6c^' + 3 


6/At- 3 


6/v + 3 


c/fcZ -18 


6^+6 


c/v + 3 




bcjl +12 


^ + 8 


bikl + 6 


hgik + 3 


/y - 18 


cPJ + 6 


cAt" + 3 




6^; +24 


cM - 12 


Ph + 3 


6Ai« - 6 


pa - 9 


c/^A; - 6 


/i^- - 3 




bgij + 6 


c/!;7 + 3 


/»A:; - 6 


hiP +12 


fhi" + 9 


cfhi - 3 


gi^l + 6 




c/A^ -18 


cP ^ S 


/iA:« + 3 


c/Ar» - 6 


t»ifcZ +18 


c/^ -12 


»!; - 3